Theorem vtocldf 3542

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
vtocld.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
vtocld.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
vtocld.3	⊢ (𝜑 → 𝜓)
vtocldf.4	⊢ Ⅎ𝑥𝜑
vtocldf.5	⊢ (𝜑 → Ⅎ𝑥𝐴)
vtocldf.6	⊢ (𝜑 → Ⅎ𝑥𝜒)

Assertion

Ref	Expression
vtocldf	⊢ (𝜑 → 𝜒)

Proof of Theorem vtocldf

Step	Hyp	Ref	Expression
1		vtocldf.5	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
2		vtocldf.6	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
3		vtocldf.4	. . 3 ⊢ Ⅎ𝑥𝜑
4		vtocld.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
5	4	ex 412	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
6	3, 5	alrimi 2199	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
7		vtocld.3	. . 3 ⊢ (𝜑 → 𝜓)
8	3, 7	alrimi 2199	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
9		vtocld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		vtoclgft 3536	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴 ∈ 𝑉) → 𝜒)
11	1, 2, 6, 8, 9, 10	syl221anc 1379	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1532   = wceq 1534  Ⅎwnf 1778   ∈ wcel 2099  Ⅎwnfc 2878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-v 3471

This theorem is referenced by:  vtocldOLD  3544  iota2df  6529  riotasv2d  38418