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Theorem syl5eqelr 2438
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
syl5eqelr.1 B = A
syl5eqelr.2 (φB C)
Assertion
Ref Expression
syl5eqelr (φA C)

Proof of Theorem syl5eqelr
StepHypRef Expression
1 syl5eqelr.1 . . 3 B = A
21eqcomi 2357 . 2 A = B
3 syl5eqelr.2 . 2 (φB C)
42, 3syl5eqel 2437 1 (φA C)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1642   wcel 1710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349
This theorem is referenced by:  xpkexg  4288  pw1exb  4326  nncaddccl  4419  0cminle  4461  vfinspsslem1  4550  opexb  4603  cnvexb  5103  epprc  5827  frds  5935  ovmuc  6130  ovcelem1  6171  ce2t  6235  addccan2nclem2  6264  fnfreclem1  6317  elscan  6330
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