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

Theorem cdeqi 2814
Description: Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqi.1 (𝑥 = 𝑦𝜑)
Assertion
Ref Expression
cdeqi CondEq(𝑥 = 𝑦𝜑)

Proof of Theorem cdeqi
StepHypRef Expression
1 cdeqi.1 . 2 (𝑥 = 𝑦𝜑)
2 df-cdeq 2813 . 2 (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
31, 2mpbir 144 1 CondEq(𝑥 = 𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  CondEqwcdeq 2812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-cdeq 2813
This theorem is referenced by:  cdeqth  2816  cdeqnot  2817  cdeqal  2818  cdeqab  2819  cdeqim  2822  cdeqcv  2823  cdeqeq  2824  cdeqel  2825
  Copyright terms: Public domain W3C validator