Theorem eleq1a 2209

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 2200	. 2 |- ( C = A -> ( C e. B <-> A e. B ) )
2	1	biimprcd 159	1 |- ( A e. B -> ( C = A -> C e. B ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133

This theorem is referenced by:  elex22  2696  elex2  2697  reu6  2868  disjne  3411  ssimaex  5475  fnex  5635  f1ocnv2d  5967  tfrlem8  6208  eroprf  6515  ac6sfi  6785  recclnq  7193  prnmaddl  7291  renegcl  8016  nn0ind-raph  9161  iccid  9701  opnneiid  12322  metrest  12664  coseq0negpitopi  12906  bj-nn0suc  13151  bj-inf2vnlem2  13158  bj-nn0sucALT  13165