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Theorem eqrd 4002
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 2207 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
4 eqrd.1 . . 3 𝑥𝐴
5 eqrd.2 . . 3 𝑥𝐵
64, 5cleqf 2935 . 2 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
73, 6sylibr 233 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540   = wceq 1542  wnf 1786  wcel 2107  wnfc 2884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-cleq 2725  df-clel 2811  df-nfc 2886
This theorem is referenced by:  eqri  4003  sniota  6535  dissnlocfin  23033  imasnopn  23194  imasncld  23195  imasncls  23196  blval2  24071  eqrrabd  31741  fimarab  31868  ofpreima  31890  zarcls  32854  ordtconnlem1  32904  qqhval2  32962  reprdifc  33639  topdifinfindis  36227  icorempo  36232  isbasisrelowllem1  36236  isbasisrelowllem2  36237  sticksstones11  40972  areaquad  41965  rfcnpre1  43703  rfcnpre2  43715  preimagelt  45415  preimalegt  45416
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