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Theorem eqvincf 2663
 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 2661 . 2 (A = By(y = A y = B))
3 eqvincf.1 . . . . 5 xA
43nfeq2 2186 . . . 4 x y = A
5 eqvincf.2 . . . . 5 xB
65nfeq2 2186 . . . 4 x y = B
74, 6nfan 1454 . . 3 x(y = A y = B)
8 nfv 1418 . . 3 y(x = A x = B)
9 eqeq1 2043 . . . 4 (y = x → (y = Ax = A))
10 eqeq1 2043 . . . 4 (y = x → (y = Bx = B))
119, 10anbi12d 442 . . 3 (y = x → ((y = A y = B) ↔ (x = A x = B)))
127, 8, 11cbvex 1636 . 2 (y(y = A y = B) ↔ x(x = A x = B))
132, 12bitri 173 1 (A = Bx(x = A x = B))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Ⅎwnfc 2162  Vcvv 2551 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553 This theorem is referenced by: (None)
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