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Theorem eqri 3995
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 1798 . . 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 3994 . 2 (⊤ → 𝐴 = 𝐵)
76mptru 1540 1 𝐴 = 𝐵
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wtru 1534  wcel 2098  wnfc 2875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2716  df-clel 2802  df-nfc 2877
This theorem is referenced by:  rnep  5917  difrab2  32210  esum2dlem  33582  eulerpartlemn  33872
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