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Theorem sbceqal 3639
Description: Set theory version of sbeqal1 39124. (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 3600 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵)))
2 sbcimg 3629 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)))
3 eqid 2771 . . . . 5 𝐴 = 𝐴
4 eqsbc3 3627 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
53, 4mpbiri 248 . . . 4 (𝐴𝑉[𝐴 / 𝑥]𝑥 = 𝐴)
6 pm5.5 350 . . . 4 ([𝐴 / 𝑥]𝑥 = 𝐴 → (([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
75, 6syl 17 . . 3 (𝐴𝑉 → (([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵) ↔ [𝐴 / 𝑥]𝑥 = 𝐵))
8 eqsbc3 3627 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
92, 7, 83bitrd 294 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
101, 9sylibd 229 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629   = wceq 1631  wcel 2145  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  sbeqalb  3640
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