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Theorem eqrd 2963
Description: Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
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.1 . . 3 𝑥𝐴
3 eqrd.2 . . 3 𝑥𝐵
4 eqrd.3 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐵))
54biimpd 132 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5ssrd 2950 . 2 (𝜑𝐴𝐵)
74biimprd 147 . . 3 (𝜑 → (𝑥𝐵𝑥𝐴))
81, 3, 2, 7ssrd 2950 . 2 (𝜑𝐵𝐴)
96, 8eqssd 2962 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wnf 1349  wcel 1393  wnfc 2165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931
This theorem is referenced by: (None)
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