Theorem jao1i 856

Description: Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)

Hypothesis

Ref	Expression
jao1i.1	⊢ (𝜓 → (𝜒 → 𝜑))

Assertion

Ref	Expression
jao1i	⊢ ((𝜑 ∨ 𝜓) → (𝜒 → 𝜑))

Proof of Theorem jao1i

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (𝜑 → (𝜒 → 𝜑))
2		jao1i.1	. 2 ⊢ (𝜓 → (𝜒 → 𝜑))
3	1, 2	jaoi 855	1 ⊢ ((𝜑 ∨ 𝜓) → (𝜒 → 𝜑))

Syntax hints:   → wi 4   ∨ wo 845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-or 846

This theorem is referenced by:  pm2.64  939  pm2.82  973  sorpssint  7742  preleqg  9644  ltlen  11351  elnnnn0b  12552  znnn0nn  12709  scshwfzeqfzo  14815  nn0enne  16359  dvdsprmpweqnn  16859  dvdsprmpweqle  16860  prmirred  21405  pmatcollpw3fi1  22708  2lgsoddprmlem3  27365  sltlend  27722  prtlem14  38350