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

Theorem tbt 357
Description: A wff is equivalent to its equivalence with a truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
Hypothesis
Ref Expression
tbt.1 𝜑
Assertion
Ref Expression
tbt (𝜓 ↔ (𝜓𝜑))

Proof of Theorem tbt
StepHypRef Expression
1 tbt.1 . 2 𝜑
2 ibibr 356 . . 3 ((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))
32pm5.74ri 259 . 2 (𝜑 → (𝜓 ↔ (𝜓𝜑)))
41, 3ax-mp 5 1 (𝜓 ↔ (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 194
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 195
This theorem is referenced by:  tbtru  1484  exists1  2548  eqvf  3176  reu6  3361  vprc  4719  elnev  37457
  Copyright terms: Public domain W3C validator