Theorem pm5.6 1004

Description: Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.)

Assertion

Ref	Expression
pm5.6	⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Proof of Theorem pm5.6

Step	Hyp	Ref	Expression
1		impexp 450	. 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
2		df-or 849	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒))
3	2	imbi2i 336	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
4	1, 3	bitr4i 278	1 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848

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-an 396  df-or 849

This theorem is referenced by:  ssundif  4488  brdom3  10568  grothprim  10874  eliccelico  32779  elicoelioo  32780  ballotlemfc0  34495  ballotlemfcc  34496  elicc3  36318  faosnf0.11b  43440  ifpidg  43504  dfsucon  43536  icccncfext  45902