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Theorem eleq1a 2249
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
StepHypRef Expression
1 eleq1 2240 . 2  |-  ( C  =  A  ->  ( C  e.  B  <->  A  e.  B ) )
21biimprcd 160 1  |-  ( A  e.  B  ->  ( C  =  A  ->  C  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148
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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173
This theorem is referenced by:  elex22  2754  elex2  2755  reu6  2928  disjne  3478  ssimaex  5579  fnex  5740  f1ocnv2d  6077  mpoexw  6216  tfrlem8  6321  eroprf  6630  ac6sfi  6900  recclnq  7393  prnmaddl  7491  renegcl  8220  nn0ind-raph  9372  iccid  9927  4sqlem1  12388  4sqlem4  12392  lssvneln0  13464  lss1d  13475  opnneiid  13703  metrest  14045  coseq0negpitopi  14296  bj-nn0suc  14755  bj-inf2vnlem2  14762  bj-nn0sucALT  14769
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