Theorem nfsbxyt 1959

Description: Closed form of nfsbxy 1958. (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 1520	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		nfs1v 1955	. . . . 5 ⊢ Ⅎ𝑧[𝑦 / 𝑧]𝜑
3		drsb1 1810	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	drnf2 1745	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
5	2, 4	mpbii 148	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
6	5	a1d 22	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
7		a16nf 1877	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8	7	a1d 22	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
9		df-nf 1472	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
10	9	albii 1481	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11		sb5 1899	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
12		nfa1 1552	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦
13		nfa1 1552	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑
14	12, 13	nfan 1576	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑)
15		sp 1522	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
16	15	adantr 276	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧 𝑥 = 𝑦)
17		sp 1522	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑)
18	17	adantl 277	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧𝜑)
19	16, 18	nfand 1579	. . . . . . . 8 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑))
20	14, 19	nfexd 1772	. . . . . . 7 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
21	11, 20	nfxfrd 1486	. . . . . 6 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
22	21	ex 115	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
23	10, 22	sylbir 135	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
24	8, 23	jaoi 717	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
25	6, 24	jaoi 717	. 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 104   ∨ wo 709  ∀wal 1362  Ⅎwnf 1471  ∃wex 1503  [wsb 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  nfsbt  1992