Theorem nfsbxyt 1936

Description: Closed form of nfsbxy 1935. (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 1502	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		nfs1v 1932	. . . . 5 ⊢ Ⅎ𝑧[𝑦 / 𝑧]𝜑
3		drsb1 1792	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	drnf2 1727	. . . . 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 1859	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8	7	a1d 22	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
9		df-nf 1454	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
10	9	albii 1463	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11		sb5 1880	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
12		nfa1 1534	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦
13		nfa1 1534	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑
14	12, 13	nfan 1558	. . . . . . . 8 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑)
15		sp 1504	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
16	15	adantr 274	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧 𝑥 = 𝑦)
17		sp 1504	. . . . . . . . . 10 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑)
18	17	adantl 275	. . . . . . . . 9 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧𝜑)
19	16, 18	nfand 1561	. . . . . . . 8 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑))
20	14, 19	nfexd 1754	. . . . . . 7 ⊢ ((∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑥Ⅎ𝑧𝜑) → Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
21	11, 20	nfxfrd 1468	. . . . . 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 711	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
25	6, 24	jaoi 711	. 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 703  ∀wal 1346  Ⅎwnf 1453  ∃wex 1485  [wsb 1755

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  nfsbt  1969