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Theorem eqvincf 2642
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 2640 . 2 (A = By(y = A y = B))
3 eqvincf.1 . . . . 5 xA
43nfeq2 2167 . . . 4 x y = A
5 eqvincf.2 . . . . 5 xB
65nfeq2 2167 . . . 4 x y = B
74, 6nfan 1435 . . 3 x(y = A y = B)
8 nfv 1398 . . 3 y(x = A x = B)
9 eqeq1 2024 . . . 4 (y = x → (y = Ax = A))
10 eqeq1 2024 . . . 4 (y = x → (y = Bx = B))
119, 10anbi12d 445 . . 3 (y = x → ((y = A y = B) ↔ (x = A x = B)))
127, 8, 11cbvex 1617 . 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 1226  wex 1358   wcel 1370  wnfc 2143  Vcvv 2531
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533
This theorem is referenced by: (None)
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