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Theorem eqri 3998
Description: Infer equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 7-Oct-2017.)
Hypotheses
Ref Expression
eqri.1 𝑥𝐴
eqri.2 𝑥𝐵
eqri.3 (𝑥𝐴𝑥𝐵)
Assertion
Ref Expression
eqri 𝐴 = 𝐵

Proof of Theorem eqri
StepHypRef Expression
1 nftru 1799 . . 3 𝑥
2 eqri.1 . . 3 𝑥𝐴
3 eqri.2 . . 3 𝑥𝐵
4 eqri.3 . . . 4 (𝑥𝐴𝑥𝐵)
54a1i 11 . . 3 (⊤ → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5eqrd 3997 . 2 (⊤ → 𝐴 = 𝐵)
76mptru 1541 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wtru 1535  wcel 2099  wnfc 2879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-cleq 2720  df-clel 2806  df-nfc 2881
This theorem is referenced by:  rnep  5923  difrab2  32289  esum2dlem  33705  eulerpartlemn  33995
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