Theorem eleq1a 2150

Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Assertion

Ref	Expression
eleq1a	|- ( A e. B -> ( C = A -> C e. B ) )

Proof of Theorem eleq1a

Step	Hyp	Ref	Expression
1		eleq1 2141	. 2 |- ( C = A -> ( C e. B <-> A e. B ) )
2	1	biimprcd 158	1 |- ( A e. B -> ( C = A -> C e. B ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  elex22  2614  elex2  2615  reu6  2781  disjne  3297  ssimaex  5255  fnex  5404  f1ocnv2d  5724  tfrlem8  5957  eroprf  6222  ac6sfi  6379  recclnq  6582  prnmaddl  6680  renegcl  7369  nn0ind-raph  8464  iccid  8948  bj-nn0suc  10759  bj-inf2vnlem2  10766  bj-nn0sucALT  10773