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Theorem cdeqel 2825
 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 2815 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
3 cdeqeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
43cdeqri 2815 . . 3 (𝑥 = 𝑦𝐶 = 𝐷)
52, 4eleq12d 2155 . 2 (𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
65cdeqi 2814 1 CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   ↔ wb 103   = wceq 1287   ∈ wcel 1436  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-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067 This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081  df-cdeq 2813 This theorem is referenced by:  nfccdeq  2827
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