ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3eltr4g Unicode version

Theorem 3eltr4g 2123
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
3eltr4g.1  |-  ( ph  ->  A  e.  B )
3eltr4g.2  |-  C  =  A
3eltr4g.3  |-  D  =  B
Assertion
Ref Expression
3eltr4g  |-  ( ph  ->  C  e.  D )

Proof of Theorem 3eltr4g
StepHypRef Expression
1 3eltr4g.1 . 2  |-  ( ph  ->  A  e.  B )
2 3eltr4g.2 . . 3  |-  C  =  A
3 3eltr4g.3 . . 3  |-  D  =  B
42, 3eleq12i 2105 . 2  |-  ( C  e.  D  <->  A  e.  B )
51, 4sylibr 137 1  |-  ( ph  ->  C  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036
This theorem is referenced by:  riotacl2  5481
  Copyright terms: Public domain W3C validator