ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl5eqelr GIF version

Theorem syl5eqelr 2141
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eqelr.1 𝐵 = 𝐴
syl5eqelr.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
syl5eqelr (𝜑𝐴𝐶)

Proof of Theorem syl5eqelr
StepHypRef Expression
1 syl5eqelr.1 . . 3 𝐵 = 𝐴
21eqcomi 2060 . 2 𝐴 = 𝐵
3 syl5eqelr.2 . 2 (𝜑𝐵𝐶)
42, 3syl5eqel 2140 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052
This theorem is referenced by:  dmrnssfld  4623  cnvexg  4883  opabbrex  5577  offval  5747  resfunexgALT  5765  abrexexg  5773  abrexex2g  5775  opabex3d  5776  nqprlu  6703  iccshftr  8963  iccshftl  8965  iccdil  8967  icccntr  8969
  Copyright terms: Public domain W3C validator