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Theorem dvelimh 2318
Description: Version of dvelim 2319 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 2009 . . 3 𝑥𝜑
3 dvelimh.2 . . . 4 (𝜓 → ∀𝑧𝜓)
43nf5i 2009 . . 3 𝑧𝜓
5 dvelimh.3 . . 3 (𝑧 = 𝑦 → (𝜑𝜓))
62, 4, 5dvelimf 2316 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
76nf5rd 2051 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by:  dvelim  2319  dveeq1-o16  33037  dveel2ALT  33040  ax6e2nd  37593  ax6e2ndVD  37964  ax6e2ndALT  37986
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