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Theorem eqrd 2957
 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 2944 . 2 (φAB)
74biimprd 147 . . 3 (φ → (x Bx A))
81, 3, 2, 7ssrd 2944 . 2 (φBA)
96, 8eqssd 2956 1 (φA = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  Ⅎwnfc 2162 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925 This theorem is referenced by: (None)
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