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Theorem cdeqnot 2814
Description: Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqnot.1 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cdeqnot CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))

Proof of Theorem cdeqnot
StepHypRef Expression
1 cdeqnot.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
21cdeqri 2812 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
32notbid 625 . 2 (𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
43cdeqi 2811 1 CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  CondEqwcdeq 2809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-cdeq 2810
This theorem is referenced by: (None)
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