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Theorem eqssd 2962
Description: Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
Hypotheses
Ref Expression
eqssd.1 (𝜑𝐴𝐵)
eqssd.2 (𝜑𝐵𝐴)
Assertion
Ref Expression
eqssd (𝜑𝐴 = 𝐵)

Proof of Theorem eqssd
StepHypRef Expression
1 eqssd.1 . 2 (𝜑𝐴𝐵)
2 eqssd.2 . 2 (𝜑𝐵𝐴)
3 eqss 2960 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
41, 2, 3sylanbrc 394 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wss 2917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931
This theorem is referenced by:  eqrd  2963  unissel  3609  intmin  3635  int0el  3645  dmcosseq  4603  relfld  4846  imadif  4979  imain  4981  fimacnv  5296  fo2ndf  5848  tposeq  5862  tfrlemibfn  5942  tfrlemi14d  5947  nndifsnid  6080  fidifsnid  6332  fisbth  6340  addnqpr  6657  mulnqpr  6673  distrprg  6684  ltexpri  6709  addcanprg  6712  recexprlemex  6733  aptipr  6737  cauappcvgprlemladd  6754  fzopth  8922  fzosplit  9031  fzouzsplit  9033  frecuzrdgfn  9172  findset  10044
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