Theorem orcanai 940

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 736	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜒))
3	2	imp 124	1 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  fsumsplit  12193  pcgcd  13131  ballotfilem2  13280  lgsdir2  16323