MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eleq12i Structured version   Unicode version

Theorem eleq12i 2501
Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
Hypotheses
Ref Expression
eleq1i.1  |-  A  =  B
eleq12i.2  |-  C  =  D
Assertion
Ref Expression
eleq12i  |-  ( A  e.  C  <->  B  e.  D )

Proof of Theorem eleq12i
StepHypRef Expression
1 eleq12i.2 . . 3  |-  C  =  D
21eleq2i 2500 . 2  |-  ( A  e.  C  <->  A  e.  D )
3 eleq1i.1 . . 3  |-  A  =  B
43eleq1i 2499 . 2  |-  ( A  e.  D  <->  B  e.  D )
52, 4bitri 241 1  |-  ( A  e.  C  <->  B  e.  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725
This theorem is referenced by:  3eltr3g  2518  3eltr4g  2519  sbcel12g  3266  bnj98  29238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2429  df-clel 2432
  Copyright terms: Public domain W3C validator