MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3eltr3g Unicode version

Theorem 3eltr3g 2486
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr3g.1  |-  ( ph  ->  A  e.  B )
3eltr3g.2  |-  A  =  C
3eltr3g.3  |-  B  =  D
Assertion
Ref Expression
3eltr3g  |-  ( ph  ->  C  e.  D )

Proof of Theorem 3eltr3g
StepHypRef Expression
1 3eltr3g.1 . 2  |-  ( ph  ->  A  e.  B )
2 3eltr3g.2 . . 3  |-  A  =  C
3 3eltr3g.3 . . 3  |-  B  =  D
42, 3eleq12i 2469 . 2  |-  ( A  e.  B  <->  C  e.  D )
51, 4sylib 189 1  |-  ( ph  ->  C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721
This theorem is referenced by:  rankelpr  7755  isf34lem7  8215  rmulccn  24267  xrge0mulc1cn  24280  esumpfinvallem  24417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2397  df-clel 2400
  Copyright terms: Public domain W3C validator