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Theorem eleq1a 2236
Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
Assertion
Ref Expression
eleq1a (𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))

Proof of Theorem eleq1a
StepHypRef Expression
1 eleq1 2227 . 2 (𝐶 = 𝐴 → (𝐶𝐵𝐴𝐵))
21biimprcd 159 1 (𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1342  wcel 2135
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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-cleq 2157  df-clel 2160
This theorem is referenced by:  elex22  2739  elex2  2740  reu6  2913  disjne  3460  ssimaex  5544  fnex  5704  f1ocnv2d  6039  tfrlem8  6280  eroprf  6588  ac6sfi  6858  recclnq  7327  prnmaddl  7425  renegcl  8153  nn0ind-raph  9302  iccid  9855  4sqlem1  12312  4sqlem4  12316  opnneiid  12762  metrest  13104  coseq0negpitopi  13355  bj-nn0suc  13739  bj-inf2vnlem2  13746  bj-nn0sucALT  13753
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