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Theorem eqrd 3119
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 143 . . 3 (𝜑 → (𝑥𝐴𝑥𝐵))
61, 2, 3, 5ssrd 3106 . 2 (𝜑𝐴𝐵)
74biimprd 157 . . 3 (𝜑 → (𝑥𝐵𝑥𝐴))
81, 3, 2, 7ssrd 3106 . 2 (𝜑𝐵𝐴)
96, 8eqssd 3118 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  wnf 1437  wcel 1481  wnfc 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3081  df-ss 3088
This theorem is referenced by:  dfss4st  3313  imasnopn  12505
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