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Theorem eqrd 4001
Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) (Proof shortened by BJ, 1-Dec-2021.)
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.3 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
31, 2alrimi 2206 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4 eqrd.1 . . 3 𝑥𝐴
5 eqrd.2 . . 3 𝑥𝐵
64, 5cleqf 2934 . 2 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
73, 6sylibr 233 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wnf 1785  wcel 2106  wnfc 2883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-cleq 2724  df-clel 2810  df-nfc 2885
This theorem is referenced by:  eqri  4002  sniota  6534  dissnlocfin  23032  imasnopn  23193  imasncld  23194  imasncls  23195  blval2  24070  eqrrabd  31736  fimarab  31863  ofpreima  31885  zarcls  32849  ordtconnlem1  32899  qqhval2  32957  reprdifc  33634  topdifinfindis  36222  icorempo  36227  isbasisrelowllem1  36231  isbasisrelowllem2  36232  sticksstones11  40967  areaquad  41955  rfcnpre1  43693  rfcnpre2  43705  preimagelt  45405  preimalegt  45406
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