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Theorem eqri 3984
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 1796 . . 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 3983 . 2 (⊤ → 𝐴 = 𝐵)
76mptru 1535 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wtru 1529  wcel 2105  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-cleq 2811  df-clel 2890  df-nfc 2960
This theorem is referenced by:  rnep  5790  difrab2  30188  esum2dlem  31250  eulerpartlemn  31538
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