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Theorem sbceqal 3097
 Description: Set theory version of sbeqal1 in set.mm. (Contributed by Andrew Salmon, 28-Jun-2011.)
Assertion
Ref Expression
sbceqal (A V → (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 3058 . 2 (A V → (x(x = Ax = B) → [̣A / x]̣(x = Ax = B)))
2 sbcimg 3087 . . 3 (A V → ([̣A / x]̣(x = Ax = B) ↔ ([̣A / xx = A → [̣A / xx = B)))
3 eqid 2353 . . . . 5 A = A
4 eqsbc3 3085 . . . . 5 (A V → ([̣A / xx = AA = A))
53, 4mpbiri 224 . . . 4 (A V → [̣A / xx = A)
6 pm5.5 326 . . . 4 ([̣A / xx = A → (([̣A / xx = A → [̣A / xx = B) ↔ [̣A / xx = B))
75, 6syl 15 . . 3 (A V → (([̣A / xx = A → [̣A / xx = B) ↔ [̣A / xx = B))
8 eqsbc3 3085 . . 3 (A V → ([̣A / xx = BA = B))
92, 7, 83bitrd 270 . 2 (A V → ([̣A / x]̣(x = Ax = B) ↔ A = B))
101, 9sylibd 205 1 (A V → (x(x = Ax = B) → A = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbeqalb  3098
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