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Theorem sbeqalb 3098
 Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
Assertion
Ref Expression
sbeqalb (A V → ((x(φx = A) x(φx = B)) → A = B))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem sbeqalb
StepHypRef Expression
1 bibi1 317 . . . . 5 ((φx = A) → ((φx = B) ↔ (x = Ax = B)))
21biimpa 470 . . . 4 (((φx = A) (φx = B)) → (x = Ax = B))
32biimpd 198 . . 3 (((φx = A) (φx = B)) → (x = Ax = B))
43alanimi 1562 . 2 ((x(φx = A) x(φx = B)) → x(x = Ax = B))
5 sbceqal 3097 . 2 (A V → (x(x = Ax = B) → A = B))
64, 5syl5 28 1 (A V → ((x(φx = A) x(φx = B)) → A = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710 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:  iotaval  4350
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