Theorem nfsb4t 2496

Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 2497). Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)

Assertion

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

Proof of Theorem nfsb4t

Step	Hyp	Ref	Expression
1		sbequ12 2241	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
2	1	sps 2176	. . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	2	drnf2 2441	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
4	3	biimpd 228	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
5	4	spsd 2178	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
6	5	impcom 406	. . 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 2149	. . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑
9	8	nfal 2314	. . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑
10		nfnae 2431	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
11	9, 10	nfan 1900	. . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
12		nfa1 2146	. . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑
13		nfnae 2431	. . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
14	12, 13	nfan 1900	. . 3 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
15		sp 2174	. . . 4 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑)
16	15	adantr 479	. . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑)
17		nfsb2 2480	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
18	17	adantl 480	. . 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 2446	. 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
21	7, 20	pm2.61dan 809	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1537  Ⅎwnf 1783  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-11 2152  ax-12 2169  ax-13 2369

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066

This theorem is referenced by:  nfsb4  2497  nfsbd  2519