Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm5.6 Structured version   Visualization version   GIF version

Theorem pm5.6 950
 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 462 . 2 (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓𝜒)))
2 df-or 385 . . 3 ((𝜓𝜒) ↔ (¬ 𝜓𝜒))
32imbi2i 326 . 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   ∨ wo 383   ∧ wa 384 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-or 385  df-an 386 This theorem is referenced by:  ssundif  4026  brdom3  9297  grothprim  9603  eliccelico  29395  elicoelioo  29396  ballotlemfc0  30347  ballotlemfcc  30348  elicc3  31974  ifpidg  37338  icccncfext  39421
 Copyright terms: Public domain W3C validator