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Theorem eqrd 3041
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 142 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5ssrd 3028 . 2 (𝜑𝐴𝐵)
74biimprd 156 . . 3 (𝜑 → (𝑥𝐵𝑥𝐴))
81, 3, 2, 7ssrd 3028 . 2 (𝜑𝐵𝐴)
96, 8eqssd 3040 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3003  df-ss 3010
This theorem is referenced by:  dfss4st  3230
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