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Theorem cdeqeq 3688
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 3679 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
3 cdeqeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
43cdeqri 3679 . . 3 (𝑥 = 𝑦𝐶 = 𝐷)
52, 4eqeq12d 2753 . 2 (𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))
65cdeqi 3678 1 CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  CondEqwcdeq 3676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2729  df-cdeq 3677
This theorem is referenced by: (None)
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