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Theorem sbceqal 2870
Description: A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
Assertion
Ref Expression
sbceqal (𝐴𝑉 → (∀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 2827 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵)))
2 sbcimg 2856 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)))
3 eqid 2082 . . . . 5 𝐴 = 𝐴
4 eqsbc3 2854 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
53, 4mpbiri 166 . . . 4 (𝐴𝑉[𝐴 / 𝑥]𝑥 = 𝐴)
6 pm5.5 240 . . . 4 ([𝐴 / 𝑥]𝑥 = 𝐴 → (([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
75, 6syl 14 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
8 eqsbc3 2854 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
92, 7, 83bitrd 212 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
101, 9sylibd 147 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283   = wceq 1285  wcel 1434  [wsbc 2816
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817
This theorem is referenced by:  sbeqalb  2871  snsssn  3561
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