Theorem nfsb4t 2377

Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2378). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Assertion

Ref	Expression
nfsb4t	⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))

Proof of Theorem nfsb4t

Step	Hyp	Ref	Expression
1		sbequ12 2097	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
2	1	sps 2043	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	2	drnf2 2318	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
4	3	biimpd 218	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
5	4	spsd 2045	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
6	5	impcom 445	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7	6	a1d 25	. 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
8		nfnf1 2018	. . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑
9	8	nfal 2139	. . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑
10		nfnae 2306	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
11	9, 10	nfan 1816	. . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
12		nfa1 2015	. . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑
13		nfnae 2306	. . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
14	12, 13	nfan 1816	. . 3 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
15		sp 2041	. . . 4 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑)
16	15	adantr 480	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑)
17		nfsb2 2348	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
18	17	adantl 481	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
19	1	a1i 11	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
20	11, 14, 16, 18, 19	dvelimdf 2323	. 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
21	7, 20	pm2.61dan 828	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  Ⅎwnf 1699  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by:  nfsb4  2378  nfsbd  2430