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  6107  sorpssint  7676  preleqg  9522  ltlen  11232  elnnnn0b  12443  znnn0nn  12601  scshwfzeqfzo  14747  nn0enne  16302  dvdsprmpweqnn  16811  dvdsprmpweqle  16812  prmirred  21427  pmatcollpw3fi1  22730  2lgsoddprmlem3  27379  sltlend  27737  prtlem14  39073