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Theorem eleq1a 2189
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 2180 . 2 (𝐶 = 𝐴 → (𝐶𝐵𝐴𝐵))
21biimprcd 159 1 (𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113
This theorem is referenced by:  elex22  2675  elex2  2676  reu6  2846  disjne  3386  ssimaex  5450  fnex  5610  f1ocnv2d  5942  tfrlem8  6183  eroprf  6490  ac6sfi  6760  recclnq  7168  prnmaddl  7266  renegcl  7991  nn0ind-raph  9136  iccid  9676  opnneiid  12260  metrest  12602  coseq0negpitopi  12844  bj-nn0suc  13089  bj-inf2vnlem2  13096  bj-nn0sucALT  13103
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