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Theorem pm5.6 992
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 → (𝜓𝜒)))

Proof of Theorem pm5.6
StepHypRef Expression
1 impexp 437 . 2 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓𝜒)))
2 df-or 827 . . 3 ((𝜓𝜒) ↔ (¬ 𝜓𝜒))
32imbi2i 325 . 2 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜓𝜒)))
41, 3bitr4i 267 1 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 826
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 197  df-an 383  df-or 827
This theorem is referenced by:  ssundif  4192  brdom3  9551  grothprim  9857  eliccelico  29873  elicoelioo  29874  ballotlemfc0  30888  ballotlemfcc  30889  elicc3  32642  ifpidg  38355  icccncfext  40612
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