Theorem jao1i 854

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 853	1 ⊢ ((𝜑 ∨ 𝜓) → (𝜒 → 𝜑))

Syntax hints:   → wi 4   ∨ wo 843

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 209  df-or 844

This theorem is referenced by:  pm2.64  938  pm2.82  972  sorpssint  7453  preleqg  9072  ltlen  10735  elnnnn0b  11935  znnn0nn  12088  scshwfzeqfzo  14182  nn0enne  15722  dvdsprmpweqnn  16215  dvdsprmpweqle  16216  prmirred  20636  pmatcollpw3fi1  21390  2lgsoddprmlem3  25984  prtlem14  36004