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Theorem eqelssd 3211
Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)
Hypotheses
Ref Expression
eqelssd.1 (𝜑𝐴𝐵)
eqelssd.2 ((𝜑𝑥𝐵) → 𝑥𝐴)
Assertion
Ref Expression
eqelssd (𝜑𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem eqelssd
StepHypRef Expression
1 eqelssd.1 . 2 (𝜑𝐴𝐵)
2 eqelssd.2 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐴)
32ex 115 . . 3 (𝜑 → (𝑥𝐵𝑥𝐴))
43ssrdv 3198 . 2 (𝜑𝐵𝐴)
51, 4eqssd 3209 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1372  wcel 2175  wss 3165
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178
This theorem is referenced by:  fiuni  7079  ennnfonelemrn  12761  ennnfonelemdm  12762  unirnblps  14865  unirnbl  14866  dvidlemap  15134  dvidrelem  15135  dvidsslem  15136  dviaddf  15148  dvimulf  15149  dvcj  15152  dvrecap  15156
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