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Theorem dvelimhw 2352
Description: Proof of dvelimh 2500 without using ax-13 2422 but with additional distinct variable conditions. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.)
Hypotheses
Ref Expression
dvelimhw.1 (𝜑 → ∀𝑥𝜑)
dvelimhw.2 (𝜓 → ∀𝑧𝜓)
dvelimhw.3 (𝑧 = 𝑦 → (𝜑𝜓))
dvelimhw.4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Assertion
Ref Expression
dvelimhw (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem dvelimhw
StepHypRef Expression
1 nfv 2005 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
2 equcom 2115 . . . . . 6 (𝑧 = 𝑦𝑦 = 𝑧)
3 nfna1 2197 . . . . . . 7 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 dvelimhw.4 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
53, 4nf5d 2295 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
62, 5nfxfrd 1939 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
7 dvelimhw.1 . . . . . . 7 (𝜑 → ∀𝑥𝜑)
87nf5i 2191 . . . . . 6 𝑥𝜑
98a1i 11 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
106, 9nfimd 1983 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
111, 10nfald 2341 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑))
12 dvelimhw.2 . . . . 5 (𝜓 → ∀𝑧𝜓)
13 dvelimhw.3 . . . . 5 (𝑧 = 𝑦 → (𝜑𝜓))
1412, 13equsalhw 2300 . . . 4 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
1514nfbii 1937 . . 3 (Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑) ↔ Ⅎ𝑥𝜓)
1611, 15sylib 209 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
1716nf5rd 2232 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wal 1635  wnf 1863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-11 2202  ax-12 2215
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
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