Theorem nfsbxy 1859

Description: Similar to hbsb 1864 but with an extra distinct variable constraint, on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
nfsbxy.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsbxy	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

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

Proof of Theorem nfsbxy

Step	Hyp	Ref	Expression
1		ax-bndl 1439	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		nfs1v 1856	. . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑧]𝜑
3		drsb1 1720	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	drnf2 1662	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
5	2, 4	mpbii 146	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
6		a16nf 1787	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7		df-nf 1390	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
8	7	albii 1399	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9		sb5 1808	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10		nfa1 1474	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦
11		sp 1441	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
12		nfsbxy.1	. . . . . . . . 9 ⊢ Ⅎ𝑧𝜑
13	12	a1i 9	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
14	11, 13	nfand 1500	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑))
15	10, 14	nfexd 1684	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
16	9, 15	nfxfrd 1404	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
17	8, 16	sylbir 133	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
18	6, 17	jaoi 668	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
19	5, 18	jaoi 668	. 2 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
20	1, 19	ax-mp 7	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 661  ∀wal 1282  Ⅎwnf 1389  ∃wex 1421  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  nfsb  1863  sbalyz  1916  opelopabsb  4015  bezoutlemmain  10387