Theorem eleq1a 2265

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 2256	. 2 |- ( C = A -> ( C e. B <-> A e. B ) )
2	1	biimprcd 160	1 |- ( A e. B -> ( C = A -> C e. B ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  elex22  2775  elex2  2776  reu6  2950  disjne  3501  ssimaex  5619  fnex  5781  f1ocnv2d  6124  mpoexw  6268  tfrlem8  6373  eroprf  6684  ac6sfi  6956  recclnq  7454  prnmaddl  7552  mpomulf  8011  renegcl  8282  nn0ind-raph  9437  iccid  9994  4sqlem1  12529  4sqlem4  12533  4sqlem11  12542  lssvneln0  13872  lss1d  13882  lspsn  13915  rnglidlmmgm  13995  opnneiid  14343  metrest  14685  coseq0negpitopi  15012  bj-nn0suc  15526  bj-inf2vnlem2  15533  bj-nn0sucALT  15540