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Theorem eqrd 2931
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 → (x Ax B))
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 → (x Ax B))
54biimpd 132 . . 3 (φ → (x Ax B))
61, 2, 3, 5ssrd 2918 . 2 (φAB)
74biimprd 147 . . 3 (φ → (x Bx A))
81, 3, 2, 7ssrd 2918 . 2 (φBA)
96, 8eqssd 2930 1 (φA = B)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1223  wnf 1322   wcel 1366  wnfc 2138
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-in 2892  df-ss 2899
This theorem is referenced by: (None)
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