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Theorem cdeqeq 2824
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 2815 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
3 cdeqeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
43cdeqri 2815 . . 3 (𝑥 = 𝑦𝐶 = 𝐷)
52, 4eqeq12d 2099 . 2 (𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))
65cdeqi 2814 1 CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1287  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  ax-5 1379  ax-gen 1381  ax-4 1443  ax-17 1462  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-cdeq 2813
This theorem is referenced by: (None)
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