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Theorem eqssi 3209
Description: Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
Hypotheses
Ref Expression
eqssi.1 𝐴𝐵
eqssi.2 𝐵𝐴
Assertion
Ref Expression
eqssi 𝐴 = 𝐵

Proof of Theorem eqssi
StepHypRef Expression
1 eqssi.1 . 2 𝐴𝐵
2 eqssi.2 . 2 𝐵𝐴
3 eqss 3208 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
41, 2, 3mpbir2an 945 1 𝐴 = 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1373  wss 3166
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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179
This theorem is referenced by:  inv1  3497  unv  3498  undifabs  3537  intab  3914  intid  4268  find  4647  limom  4662  dmv  4894  0ima  5042  rnxpid  5117  dftpos4  6349  axaddf  7981  axmulf  7982  dfuzi  9483  unirnioo  10095  0bits  12270  4sqlem19  12732  txuni2  14728  dvef  15199  reeff1o  15245
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