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

Theorem cdeqel 2987
Description: Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypotheses
Ref Expression
cdeqeq.1  |- CondEq ( x  =  y  ->  A  =  B )
cdeqeq.2  |- CondEq ( x  =  y  ->  C  =  D )
Assertion
Ref Expression
cdeqel  |- CondEq ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )

Proof of Theorem cdeqel
StepHypRef Expression
1 cdeqeq.1 . . . 4  |- CondEq ( x  =  y  ->  A  =  B )
21cdeqri 2977 . . 3  |-  ( x  =  y  ->  A  =  B )
3 cdeqeq.2 . . . 4  |- CondEq ( x  =  y  ->  C  =  D )
43cdeqri 2977 . . 3  |-  ( x  =  y  ->  C  =  D )
52, 4eleq12d 2351 . 2  |-  ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
65cdeqi 2976 1  |- CondEq ( x  =  y  ->  ( A  e.  C  <->  B  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684  CondEqwcdeq 2974
This theorem is referenced by:  nfccdeq  2989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279  df-cdeq 2975
  Copyright terms: Public domain W3C validator