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Theorem eleq1a 2211
 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 2202 . 2 (𝐶 = 𝐴 → (𝐶𝐵𝐴𝐵))
21biimprcd 159 1 (𝐴𝐵 → (𝐶 = 𝐴𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480 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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135 This theorem is referenced by:  elex22  2701  elex2  2702  reu6  2873  disjne  3416  ssimaex  5482  fnex  5642  f1ocnv2d  5974  tfrlem8  6215  eroprf  6522  ac6sfi  6792  recclnq  7207  prnmaddl  7305  renegcl  8030  nn0ind-raph  9175  iccid  9715  opnneiid  12343  metrest  12685  coseq0negpitopi  12930  bj-nn0suc  13192  bj-inf2vnlem2  13199  bj-nn0sucALT  13206
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