Theorem jao1i 853

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 852	1 ⊢ ((𝜑 ∨ 𝜓) → (𝜒 → 𝜑))

Syntax hints:   → wi 4   ∨ wo 842

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 208  df-or 843

This theorem is referenced by:  pm2.64  936  pm2.82  970  sorpssint  7324  preleqg  8931  ltlen  10594  elnnnn0b  11795  znnn0nn  11948  scshwfzeqfzo  14028  nn0enne  15565  dvdsprmpweqnn  16054  dvdsprmpweqle  16055  prmirred  20328  pmatcollpw3fi1  21084  2lgsoddprmlem3  25676  prtlem14  35562