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Theorem dvelimh 2440
Description: Version of dvelim 2441 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.)
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 2137 . . 3 𝑥𝜑
3 dvelimh.2 . . . 4 (𝜓 → ∀𝑧𝜓)
43nf5i 2137 . . 3 𝑧𝜓
5 dvelimh.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
62, 4, 5dvelimf 2438 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
76nf5rd 2177 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823
This theorem is referenced by:  dvelim  2441  dveeq1-o16  34642  dveel2ALT  34645  ax6e2nd  39193  ax6e2ndVD  39560  ax6e2ndALT  39582
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