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Theorem eqri 29655
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 1878 . . 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 3771 . 2 (⊤ → 𝐴 = 𝐵)
76trud 1641 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wtru 1632  wcel 2145  wnfc 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-cleq 2764  df-clel 2767  df-nfc 2902
This theorem is referenced by:  difrab2  29677  esum2dlem  30494  eulerpartlemn  30783
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