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Theorem sbceqal 3685
Description: Set theory version of sbeqal1 39092. (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 3646 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵)))
2 sbcimg 3675 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)))
3 eqid 2806 . . . . 5 𝐴 = 𝐴
4 eqsbc3 3673 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
53, 4mpbiri 249 . . . 4 (𝐴𝑉[𝐴 / 𝑥]𝑥 = 𝐴)
6 pm5.5 352 . . . 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 3673 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
92, 7, 83bitrd 296 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
101, 9sylibd 230 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1635   = wceq 1637  wcel 2156  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393  df-sbc 3634
This theorem is referenced by:  sbeqalb  3686
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