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Theorem cdeqi 2894
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 2893 . 2 (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
31, 2mpbir 145 1 CondEq(𝑥 = 𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  CondEqwcdeq 2892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116  df-cdeq 2893
This theorem is referenced by:  cdeqth  2896  cdeqnot  2897  cdeqal  2898  cdeqab  2899  cdeqim  2902  cdeqcv  2903  cdeqeq  2904  cdeqel  2905
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