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Theorem cdeqel 2820
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 2810 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
3 cdeqeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
43cdeqri 2810 . . 3 (𝑥 = 𝑦𝐶 = 𝐷)
52, 4eleq12d 2153 . 2 (𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
65cdeqi 2809 1 CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1285  wcel 1434  CondEqwcdeq 2807
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079  df-cdeq 2808
This theorem is referenced by:  nfccdeq  2822
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