Theorem pm5.6 1017

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 455	. 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
2		df-or 861	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒))
3	2	imbi2i 339	. 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒)))
4	1, 3	bitr4i 281	1 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860

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 210  df-an 401  df-or 861

This theorem is referenced by:  ssundif  4444  brdom3  10500  grothprim  10807  eliccelico  33034  elicoelioo  33035  ballotlemfc0  34800  ballotlemfcc  34801  elicc3  36690  faosnf0.11b  44015  ifpidg  44079  dfsucon  44111  icccncfext  46459