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Theorem eqrd 3586
Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Hypotheses
Ref Expression
eqrd.0 𝑥𝜑
eqrd.1 𝑥𝐴
eqrd.2 𝑥𝐵
eqrd.3 (𝜑 → (𝑥𝐴𝑥𝐵))
Assertion
Ref Expression
eqrd (𝜑𝐴 = 𝐵)

Proof of Theorem eqrd
StepHypRef Expression
1 eqrd.0 . . 3 𝑥𝜑
2 eqrd.1 . . 3 𝑥𝐴
3 eqrd.2 . . 3 𝑥𝐵
4 eqrd.3 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
54biimpd 218 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5ssrd 3573 . 2 (𝜑𝐴𝐵)
74biimprd 237 . . 3 (𝜑 → (𝑥𝐵𝑥𝐴))
81, 3, 2, 7ssrd 3573 . 2 (𝜑𝐵𝐴)
96, 8eqssd 3585 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wnf 1699  wcel 1977  wnfc 2738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554
This theorem is referenced by:  sniota  5781  dissnlocfin  21090  imasnopn  21251  imasncld  21252  imasncls  21253  blval2  22125  eqri  28529  fimarab  28619  ofpreima  28642  ordtconlem1  29092  qqhval2  29148  bj-sspwpwab  32033  bj-xnex  32039  topdifinfindis  32164  icorempt2  32169  isbasisrelowllem1  32173  isbasisrelowllem2  32174  areaquad  36615  rfcnpre1  37995  rfcnpre2  38007
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