Theorem eleq1a 2301

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 2292	. 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 1395   e. wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  elex22  2815  elex2  2816  reu6  2992  disjne  3545  ssimaex  5695  fnex  5861  f1ocnv2d  6210  mpoexw  6359  tfrlem8  6464  eroprf  6775  ac6sfi  7060  recclnq  7579  prnmaddl  7677  mpomulf  8136  renegcl  8407  nn0ind-raph  9564  iccid  10121  4sqlem1  12911  4sqlem4  12915  4sqlem11  12924  lssvneln0  14337  lss1d  14347  lspsn  14380  rnglidlmmgm  14460  opnneiid  14838  metrest  15180  coseq0negpitopi  15510  bj-nn0suc  16327  bj-inf2vnlem2  16334  bj-nn0sucALT  16341