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Theorem eqvinc 2966
 Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
eqvinc.1 A V
Assertion
Ref Expression
eqvinc (A = Bx(x = A x = B))
Distinct variable groups:   x,A   x,B

Proof of Theorem eqvinc
StepHypRef Expression
1 eqvinc.1 . . . . 5 A V
21isseti 2865 . . . 4 x x = A
3 ax-1 5 . . . . . 6 (x = A → (A = Bx = A))
4 eqtr 2370 . . . . . . 7 ((x = A A = B) → x = B)
54ex 423 . . . . . 6 (x = A → (A = Bx = B))
63, 5jca 518 . . . . 5 (x = A → ((A = Bx = A) (A = Bx = B)))
76eximi 1576 . . . 4 (x x = Ax((A = Bx = A) (A = Bx = B)))
8 pm3.43 832 . . . . 5 (((A = Bx = A) (A = Bx = B)) → (A = B → (x = A x = B)))
98eximi 1576 . . . 4 (x((A = Bx = A) (A = Bx = B)) → x(A = B → (x = A x = B)))
102, 7, 9mp2b 9 . . 3 x(A = B → (x = A x = B))
111019.37aiv 1900 . 2 (A = Bx(x = A x = B))
12 eqtr2 2371 . . 3 ((x = A x = B) → A = B)
1312exlimiv 1634 . 2 (x(x = A x = B) → A = B)
1411, 13impbii 180 1 (A = Bx(x = A x = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  eqvincf  2967  preaddccan2lem1  4454  dff13  5471  nncdiv3lem1  6275
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