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Theorem eqelssd 3116
 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 114 . . 3 (𝜑 → (𝑥𝐵𝑥𝐴))
43ssrdv 3103 . 2 (𝜑𝐵𝐴)
51, 4eqssd 3114 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ⊆ wss 3071 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-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084 This theorem is referenced by:  fiuni  6866  ennnfonelemrn  11943  ennnfonelemdm  11944  unirnblps  12605  unirnbl  12606  dvidlemap  12843  dviaddf  12852  dvimulf  12853  dvcj  12856  dvrecap  12860
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