Theorem nfsbxyt 1916

Description: Closed form of nfsbxy 1915. (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 1486	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		nfs1v 1912	. . . . 5 ⊢ Ⅎ𝑧[𝑦 / 𝑧]𝜑
3		drsb1 1771	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	drnf2 1712	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
5	2, 4	mpbii 147	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
6	5	a1d 22	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
7		a16nf 1838	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8	7	a1d 22	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
9		df-nf 1437	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
10	9	albii 1446	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11		sb5 1859	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
12		nfa1 1521	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦
13		nfa1 1521	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑
14	12, 13	nfan 1544	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑)
15		sp 1488	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
16	15	adantr 274	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧 𝑥 = 𝑦)
17		sp 1488	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑)
18	17	adantl 275	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧𝜑)
19	16, 18	nfand 1547	. . . . . . . 8 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑))
20	14, 19	nfexd 1734	. . . . . . 7 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
21	11, 20	nfxfrd 1451	. . . . . 6 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
22	21	ex 114	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
23	10, 22	sylbir 134	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
24	8, 23	jaoi 705	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
25	6, 24	jaoi 705	. 2 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
26	1, 25	ax-mp 5	1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  nfsbt  1949