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Theorem sbceqal 2808
Description: A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
Assertion
Ref Expression
sbceqal (A 𝑉 → (x(x = Ax = B) → A = B))
Distinct variable groups:   x,B   x,A
Allowed substitution hint:   𝑉(x)

Proof of Theorem sbceqal
StepHypRef Expression
1 spsbc 2769 . 2 (A 𝑉 → (x(x = Ax = B) → [A / x](x = Ax = B)))
2 sbcimg 2798 . . 3 (A 𝑉 → ([A / x](x = Ax = B) ↔ ([A / x]x = A[A / x]x = B)))
3 eqid 2037 . . . . 5 A = A
4 eqsbc3 2796 . . . . 5 (A 𝑉 → ([A / x]x = AA = A))
53, 4mpbiri 157 . . . 4 (A 𝑉[A / x]x = A)
6 pm5.5 231 . . . 4 ([A / x]x = A → (([A / x]x = A[A / x]x = B) ↔ [A / x]x = B))
75, 6syl 14 . . 3 (A 𝑉 → (([A / x]x = A[A / x]x = B) ↔ [A / x]x = B))
8 eqsbc3 2796 . . 3 (A 𝑉 → ([A / x]x = BA = B))
92, 7, 83bitrd 203 . 2 (A 𝑉 → ([A / x](x = Ax = B) ↔ A = B))
101, 9sylibd 138 1 (A 𝑉 → (x(x = Ax = B) → A = B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242   wcel 1390  [wsbc 2758
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759
This theorem is referenced by:  sbeqalb  2809  snsssn  3523
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