ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl5eleqr Unicode version

Theorem syl5eleqr 2169
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eleqr.1  |-  A  e.  B
syl5eleqr.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
syl5eleqr  |-  ( ph  ->  A  e.  C )

Proof of Theorem syl5eleqr
StepHypRef Expression
1 syl5eleqr.1 . 2  |-  A  e.  B
2 syl5eleqr.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2087 . 2  |-  ( ph  ->  B  =  C )
41, 3syl5eleq 2168 1  |-  ( ph  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078
This theorem is referenced by:  rabsnt  3475  0elnn  4366  tfrexlem  5983  rdgtfr  6023  rdgruledefgg  6024  sizeinf  9802
  Copyright terms: Public domain W3C validator