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Theorem cdeqel 2760
Description: Distribute conditional equality over elementhood. (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
cdeqel CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))

Proof of Theorem cdeqel
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, 4eleq12d 2108 . 2 (𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
65cdeqi 2749 1 CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1243  wcel 1393  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-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-cdeq 2748
This theorem is referenced by:  nfccdeq  2762
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