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Theorem cdeqim 2822
Description: Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
cdeqnot.1 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
cdeqim.1 CondEq(𝑥 = 𝑦 → (𝜒𝜃))
Assertion
Ref Expression
cdeqim CondEq(𝑥 = 𝑦 → ((𝜑𝜒) ↔ (𝜓𝜃)))

Proof of Theorem cdeqim
StepHypRef Expression
1 cdeqnot.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
21cdeqri 2815 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
3 cdeqim.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜒𝜃))
43cdeqri 2815 . . 3 (𝑥 = 𝑦 → (𝜒𝜃))
52, 4imbi12d 232 . 2 (𝑥 = 𝑦 → ((𝜑𝜒) ↔ (𝜓𝜃)))
65cdeqi 2814 1 CondEq(𝑥 = 𝑦 → ((𝜑𝜒) ↔ (𝜓𝜃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  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: (None)
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