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Theorem cdeqi 3453
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 3452 . 2 (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
31, 2mpbir 221 1 CondEq(𝑥 = 𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  CondEqwcdeq 3451
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 3452
This theorem is referenced by:  cdeqth  3455  cdeqnot  3456  cdeqal  3457  cdeqab  3458  cdeqim  3461  cdeqcv  3462  cdeqeq  3463  cdeqel  3464  bj-cdeqab  32912
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