ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bibi1d Structured version   GIF version

Theorem bibi1d 222
Description: Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
imbid.1 (φ → (ψχ))
Assertion
Ref Expression
bibi1d (φ → ((ψθ) ↔ (χθ)))

Proof of Theorem bibi1d
StepHypRef Expression
1 imbid.1 . . 3 (φ → (ψχ))
21bibi2d 221 . 2 (φ → ((θψ) ↔ (θχ)))
3 bicom 128 . 2 ((ψθ) ↔ (θψ))
4 bicom 128 . 2 ((χθ) ↔ (θχ))
52, 3, 43bitr4g 212 1 (φ → ((ψθ) ↔ (χθ)))
Colors of variables: wff set class
Syntax hints:  wi 4  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:  bibi12d  224  bibi1  229  biassdc  1283  eubidh  1903  eubid  1904  axext3  2020  bm1.1  2022  eqeq1  2043  pm13.183  2675  elabgt  2678  elrab3t  2691  mob  2717  sbctt  2818  sbcabel  2833  isoeq2  5385  caovcang  5604  expap0  8919  bdsepnft  9321  bdsepnfALT  9323  strcollnft  9414  strcollnfALT  9416
  Copyright terms: Public domain W3C validator