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Theorem sbceqal 2936
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 2893 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵)))
2 sbcimg 2922 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)))
3 eqid 2117 . . . . 5 𝐴 = 𝐴
4 eqsbc3 2920 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
53, 4mpbiri 167 . . . 4 (𝐴𝑉[𝐴 / 𝑥]𝑥 = 𝐴)
6 pm5.5 241 . . . 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 2920 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
92, 7, 83bitrd 213 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
101, 9sylibd 148 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1314   = wceq 1316  wcel 1465  [wsbc 2882
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by:  sbeqalb  2937  snsssn  3658
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