Theorem eleq1a 2184

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 2175	. 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 1312   e. wcel 1461

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-17 1487  ax-ial 1495  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-cleq 2106  df-clel 2109

This theorem is referenced by:  elex22  2670  elex2  2671  reu6  2840  disjne  3380  ssimaex  5434  fnex  5594  f1ocnv2d  5926  tfrlem8  6167  eroprf  6474  ac6sfi  6743  recclnq  7142  prnmaddl  7240  renegcl  7940  nn0ind-raph  9066  iccid  9595  opnneiid  12170  metrest  12489  bj-nn0suc  12845  bj-inf2vnlem2  12852  bj-nn0sucALT  12859