Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdeqel Structured version   Visualization version   GIF version

Theorem cdeqel 3464
 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 3454 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
3 cdeqeq.2 . . . 4 CondEq(𝑥 = 𝑦𝐶 = 𝐷)
43cdeqri 3454 . . 3 (𝑥 = 𝑦𝐶 = 𝐷)
52, 4eleq12d 2724 . 2 (𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
65cdeqi 3453 1 CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1523   ∈ wcel 2030  CondEqwcdeq 3451 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-cleq 2644  df-clel 2647  df-cdeq 3452 This theorem is referenced by:  nfccdeq  3466
 Copyright terms: Public domain W3C validator