Theorem nfsb4tALT 2604

Description: Alternate version of nfsb4t 2539. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dfsb1.p5	⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Assertion

Ref	Expression
nfsb4tALT	⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))

Proof of Theorem nfsb4tALT

Step	Hyp	Ref	Expression
1		dfsb1.p5	. . . . . . . . 9 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2	1	sbequ12ALT 2581	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))
3	2	sps 2184	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜃))
4	3	drnf2 2466	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜃))
5	4	biimpd 231	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧𝜃))
6	5	spsd 2186	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜃))
7	6	impcom 410	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)
8	7	a1d 25	. 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))
9		nfnf1 2158	. . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑
10	9	nfal 2342	. . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑
11		nfnae 2456	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
12	10, 11	nfan 1900	. . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
13		nfa1 2155	. . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑
14		nfnae 2456	. . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
15	13, 14	nfan 1900	. . 3 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
16		sp 2182	. . . 4 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑)
17	16	adantr 483	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑)
18	1	nfsb2ALT 2600	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜃)
19	18	adantl 484	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜃)
20	2	a1i 11	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜑 ↔ 𝜃)))
21	12, 15, 17, 19, 20	dvelimdf 2471	. 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))
22	8, 21	pm2.61dan 811	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785

This theorem is referenced by:  nfsb4ALT  2605