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Theorem cdeqel 3398
 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 3388 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
3 cdeqeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
43cdeqri 3388 . . 3 (𝑥 = 𝑦𝐶 = 𝐷)
52, 4eleq12d 2682 . 2 (𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
65cdeqi 3387 1 CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  CondEqwcdeq 3385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-cdeq 3386 This theorem is referenced by:  nfccdeq  3400
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