Theorem orcanai 929

Description: Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)

Hypothesis

Ref	Expression
orcanai.1	⊢ (𝜑 → (𝜓 ∨ 𝜒))

Assertion

Ref	Expression
orcanai	⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)

Proof of Theorem orcanai

Step	Hyp	Ref	Expression
1		orcanai.1	. . 3 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
2	1	ord 725	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
3	2	imp 124	1 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  fsumsplit  11572  pcgcd  12498  lgsdir2  15274