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Theorem dvelimf 1997
 Description: Version of dvelimv 1939 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
dvelimf.1 xφ
dvelimf.2 zψ
dvelimf.3 (z = y → (φψ))
Assertion
Ref Expression
dvelimf x x = y → Ⅎxψ)

Proof of Theorem dvelimf
StepHypRef Expression
1 dvelimf.2 . . . 4 zψ
2 dvelimf.3 . . . 4 (z = y → (φψ))
31, 2equsal 1960 . . 3 (z(z = yφ) ↔ ψ)
43bicomi 193 . 2 (ψz(z = yφ))
5 nfnae 1956 . . 3 z ¬ x x = y
6 nfnae 1956 . . . . . 6 x ¬ x x = y
7 nfnae 1956 . . . . . 6 x ¬ x x = z
86, 7nfan 1824 . . . . 5 xx x = y ¬ x x = z)
9 ax12o 1934 . . . . . 6 x x = z → (¬ x x = y → (z = yx z = y)))
109impcom 419 . . . . 5 ((¬ x x = y ¬ x x = z) → (z = yx z = y))
118, 10nfd 1766 . . . 4 ((¬ x x = y ¬ x x = z) → Ⅎx z = y)
12 dvelimf.1 . . . . 5 xφ
1312a1i 10 . . . 4 ((¬ x x = y ¬ x x = z) → Ⅎxφ)
1411, 13nfimd 1808 . . 3 ((¬ x x = y ¬ x x = z) → Ⅎx(z = yφ))
155, 14nfald2 1972 . 2 x x = y → Ⅎxz(z = yφ))
164, 15nfxfrd 1571 1 x x = y → Ⅎxψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  dvelimnf  2017
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