Theorem jao1i 858

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 857	1 ⊢ ((𝜑 ∨ 𝜓) → (𝜒 → 𝜑))

Syntax hints:   → wi 4   ∨ wo 847

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 207  df-or 848

This theorem is referenced by:  pm2.64  943  pm2.82  977  imadifssran  6145  sorpssint  7732  preleqg  9634  ltlen  11341  elnnnn0b  12550  znnn0nn  12709  scshwfzeqfzo  14850  nn0enne  16401  dvdsprmpweqnn  16910  dvdsprmpweqle  16911  prmirred  21440  pmatcollpw3fi1  22731  2lgsoddprmlem3  27382  sltlend  27740  prtlem14  38897