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Theorem dvelimh 2450
Description: Version of dvelim 2451 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2372. Check out dvelimhw 2343 for a version requiring fewer axioms. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvelimh.1 (𝜑 → ∀𝑥𝜑)
dvelimh.2 (𝜓 → ∀𝑧𝜓)
dvelimh.3 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimh (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem dvelimh
StepHypRef Expression
1 dvelimh.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nf5i 2142 . . 3 𝑥𝜑
3 dvelimh.2 . . . 4 (𝜓 → ∀𝑧𝜓)
43nf5i 2142 . . 3 𝑧𝜓
5 dvelimh.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
62, 4, 5dvelimf 2448 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
76nf5rd 2189 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787
This theorem is referenced by:  dvelim  2451  dveeq1-o16  36950  dveel2ALT  36953  ax6e2nd  42178  ax6e2ndVD  42528  ax6e2ndALT  42550
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