Theorem nfsbxyt 1819

Description: Closed form of nfsbxy 1818. (Contributed by Jim Kingdon, 9-May-2018.)

Assertion

Ref	Expression
nfsbxyt	⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbxyt

Step	Hyp	Ref	Expression
1		ax-bndl 1399	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		nfs1v 1815	. . . . 5 ⊢ Ⅎ𝑧[𝑦 / 𝑧]𝜑
3		drsb1 1680	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	drnf2 1622	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
5	2, 4	mpbii 136	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
6	5	a1d 22	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
7		a16nf 1746	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8	7	a1d 22	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
9		df-nf 1350	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
10	9	albii 1359	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11		sb5 1767	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
12		nfa1 1434	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦
13		nfa1 1434	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑
14	12, 13	nfan 1457	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑)
15		sp 1401	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
16	15	adantr 261	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧 𝑥 = 𝑦)
17		sp 1401	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑)
18	17	adantl 262	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧𝜑)
19	16, 18	nfand 1460	. . . . . . . 8 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑))
20	14, 19	nfexd 1644	. . . . . . 7 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
21	11, 20	nfxfrd 1364	. . . . . 6 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
22	21	ex 108	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
23	10, 22	sylbir 125	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
24	8, 23	jaoi 636	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
25	6, 24	jaoi 636	. 2 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
26	1, 25	ax-mp 7	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  nfsbt  1850