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Theorem cdeqeq 2759
 Description: Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
cdeqeq.1 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
cdeqeq.2 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
cdeqeq CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem cdeqeq
StepHypRef Expression
1 cdeqeq.1 . . . 4 CondEq(𝑥 = 𝑦𝐴 = 𝐵)
21cdeqri 2750 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
3 cdeqeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
43cdeqri 2750 . . 3 (𝑥 = 𝑦𝐶 = 𝐷)
52, 4eqeq12d 2054 . 2 (𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))
65cdeqi 2749 1 CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243  CondEqwcdeq 2747 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-cdeq 2748 This theorem is referenced by: (None)
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