ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pm5.32d GIF version

Theorem pm5.32d 423
Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 29-Oct-1996.) (Revised by NM, 31-Jan-2015.)
Hypothesis
Ref Expression
pm5.32d.1 (𝜑 → (𝜓 → (𝜒𝜃)))
Assertion
Ref Expression
pm5.32d (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))

Proof of Theorem pm5.32d
StepHypRef Expression
1 pm5.32d.1 . . . 4 (𝜑 → (𝜓 → (𝜒𝜃)))
2 bi1 111 . . . 4 ((𝜒𝜃) → (𝜒𝜃))
31, 2syl6 29 . . 3 (𝜑 → (𝜓 → (𝜒𝜃)))
43imdistand 421 . 2 (𝜑 → ((𝜓𝜒) → (𝜓𝜃)))
5 bi2 121 . . . 4 ((𝜒𝜃) → (𝜃𝜒))
61, 5syl6 29 . . 3 (𝜑 → (𝜓 → (𝜃𝜒)))
76imdistand 421 . 2 (𝜑 → ((𝜓𝜃) → (𝜓𝜒)))
84, 7impbid 120 1 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  pm5.32rd  424  pm5.32da  425  pm5.32  426  anbi2d  437  cbvex2  1797  cores  4824  isoini  5457  mpt2eq123  5564  genpassl  6620  genpassu  6621  fzind  8351  btwnz  8355  elfzm11  8951
  Copyright terms: Public domain W3C validator