Theorem dvelimhw 2345

Description: Proof of dvelimh 2450 without using ax-13 2372 but with additional distinct variable conditions. (Contributed by NM, 1-Oct-2002.) (Revised 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 → (𝜑 ↔ 𝜓))
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

Step	Hyp	Ref	Expression
1		nfv 1918	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
2		equcom 2022	. . . . . 6 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
3		nfna1 2151	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4		dvelimhw.4	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
5	3, 4	nf5d 2284	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
6	2, 5	nfxfrd 1857	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
7		dvelimhw.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑥𝜑)
8	7	nf5i 2144	. . . . . 6 ⊢ Ⅎ𝑥𝜑
9	8	a1i 11	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
10	6, 9	nfimd 1898	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜑))
11	1, 10	nfald 2326	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜑))
12		dvelimhw.2	. . . . 5 ⊢ (𝜓 → ∀𝑧𝜓)
13		dvelimhw.3	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))
14	12, 13	equsalhw 2291	. . . 4 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓)
15	14	nfbii 1855	. . 3 ⊢ (Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ Ⅎ𝑥𝜓)
16	11, 15	sylib 217	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
17	16	nf5rd 2192	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)