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Theorem sbceqal 3832
Description: Class version of one implication of equvelv 2029. (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 3782 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → [𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵)))
2 sbcimg 3817 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ ([𝐴 / 𝑥]𝑥 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)))
3 eqid 2818 . . . . 5 𝐴 = 𝐴
4 eqsbc3 3814 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐴𝐴 = 𝐴))
53, 4mpbiri 259 . . . 4 (𝐴𝑉[𝐴 / 𝑥]𝑥 = 𝐴)
6 pm5.5 363 . . . 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 3814 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
92, 7, 83bitrd 306 . 2 (𝐴𝑉 → ([𝐴 / 𝑥](𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
101, 9sylibd 240 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526   = wceq 1528  wcel 2105  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-sbc 3770
This theorem is referenced by:  sbeqalb  3833
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