Theorem tfinds3 7873

Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)

Hypotheses

Ref	Expression
tfinds3.1	⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
tfinds3.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
tfinds3.3	⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
tfinds3.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
tfinds3.5	⊢ (𝜂 → 𝜓)
tfinds3.6	⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃)))
tfinds3.7	⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))

Assertion

Ref	Expression
tfinds3	⊢ (𝐴 ∈ On → (𝜂 → 𝜏))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑦   𝜒,𝑥   𝜏,𝑥   𝑥,𝑦,𝜂

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑦)   𝜃(𝑥,𝑦)   𝜏(𝑦)   𝐴(𝑦)

Proof of Theorem tfinds3

Step	Hyp	Ref	Expression
1		tfinds3.1	. . 3 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
2	1	imbi2d 339	. 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓)))
3		tfinds3.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
4	3	imbi2d 339	. 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒)))
5		tfinds3.3	. . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))
6	5	imbi2d 339	. 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃)))
7		tfinds3.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
8	7	imbi2d 339	. 2 ⊢ (𝑥 = 𝐴 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜏)))
9		tfinds3.5	. 2 ⊢ (𝜂 → 𝜓)
10		tfinds3.6	. . 3 ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃)))
11	10	a2d 29	. 2 ⊢ (𝑦 ∈ On → ((𝜂 → 𝜒) → (𝜂 → 𝜃)))
12		r19.21v 3175	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) ↔ (𝜂 → ∀𝑦 ∈ 𝑥 𝜒))
13		tfinds3.7	. . . 4 ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑)))
14	13	a2d 29	. . 3 ⊢ (Lim 𝑥 → ((𝜂 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜂 → 𝜑)))
15	12, 14	biimtrid 241	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑)))
16	2, 4, 6, 8, 9, 11, 15	tfinds 7868	1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  ∀wral 3057  ∅c0 4324  Oncon0 6372  Lim wlim 6373  suc csuc 6374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-tr 5268  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378

This theorem is referenced by:  oacl  8560  omcl  8561  oecl  8562  oawordri  8575  oaass  8586  oarec  8587  omordi  8591  omwordri  8597  odi  8604  omass  8605  oen0  8611  oewordri  8617  oeworde  8618  oeoelem  8623  omabs  8676  tfindsd  43645