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

Theorem cdeqri 3408
Description: Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqri.1 CondEq(𝑥 = 𝑦𝜑)
Assertion
Ref Expression
cdeqri (𝑥 = 𝑦𝜑)

Proof of Theorem cdeqri
StepHypRef Expression
1 cdeqri.1 . 2 CondEq(𝑥 = 𝑦𝜑)
2 df-cdeq 3406 . 2 (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
31, 2mpbi 220 1 (𝑥 = 𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  CondEqwcdeq 3405
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-cdeq 3406
This theorem is referenced by:  cdeqnot  3410  cdeqal  3411  cdeqab  3412  cdeqal1  3413  cdeqab1  3414  cdeqim  3415  cdeqeq  3417  cdeqel  3418  nfcdeq  3419  bj-cdeqab  32483
  Copyright terms: Public domain W3C validator