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Theorem cdeqim 3461
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 3454 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
3 cdeqim.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜒𝜃))
43cdeqri 3454 . . 3 (𝑥 = 𝑦 → (𝜒𝜃))
52, 4imbi12d 333 . 2 (𝑥 = 𝑦 → ((𝜑𝜒) ↔ (𝜓𝜃)))
65cdeqi 3453 1 CondEq(𝑥 = 𝑦 → ((𝜑𝜒) ↔ (𝜓𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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: (None)
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