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

Theorem pm5.32ri 436
Description: Distribution of implication over biconditional (inference rule). (Contributed by NM, 12-Mar-1995.)
Hypothesis
Ref Expression
pm5.32i.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
pm5.32ri ((𝜓𝜑) ↔ (𝜒𝜑))

Proof of Theorem pm5.32ri
StepHypRef Expression
1 pm5.32i.1 . . 3 (𝜑 → (𝜓𝜒))
21pm5.32i 435 . 2 ((𝜑𝜓) ↔ (𝜑𝜒))
3 ancom 257 . 2 ((𝜓𝜑) ↔ (𝜑𝜓))
4 ancom 257 . 2 ((𝜒𝜑) ↔ (𝜑𝜒))
52, 3, 43bitr4i 205 1 ((𝜓𝜑) ↔ (𝜒𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  anbi1i  439  pm5.36  552  pm5.61  718  oranabs  739  ceqsralt  2598  ceqsrexbv  2697  reuind  2766  rabsn  3464  dfoprab2  5579  xpsnen  6325  nn1suc  8008
  Copyright terms: Public domain W3C validator