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Theorem eqvincf 2967
 Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
eqvincf.1 xA
eqvincf.2 xB
eqvincf.3 A V
Assertion
Ref Expression
eqvincf (A = Bx(x = A x = B))

Proof of Theorem eqvincf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eqvincf.3 . . 3 A V
21eqvinc 2966 . 2 (A = By(y = A y = B))
3 eqvincf.1 . . . . 5 xA
43nfeq2 2500 . . . 4 x y = A
5 eqvincf.2 . . . . 5 xB
65nfeq2 2500 . . . 4 x y = B
74, 6nfan 1824 . . 3 x(y = A y = B)
8 nfv 1619 . . 3 y(x = A x = B)
9 eqeq1 2359 . . . 4 (y = x → (y = Ax = A))
10 eqeq1 2359 . . . 4 (y = x → (y = Bx = B))
119, 10anbi12d 691 . . 3 (y = x → ((y = A y = B) ↔ (x = A x = B)))
127, 8, 11cbvex 1985 . 2 (y(y = A y = B) ↔ x(x = A x = B))
132, 12bitri 240 1 (A = Bx(x = A x = B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859 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 This theorem is referenced by: (None)
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