Theorem nfsbxy 1961

Description: Similar to hbsb 1968 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 1523	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		nfs1v 1958	. . . 4 ⊢ Ⅎ𝑧[𝑦 / 𝑧]𝜑
3		drsb1 1813	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	drnf2 1748	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
5	2, 4	mpbii 148	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
6		a16nf 1880	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7		df-nf 1475	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
8	7	albii 1484	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9		sb5 1902	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
10		nfa1 1555	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦
11		sp 1525	. . . . . . . 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 1582	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑))
15	10, 14	nfexd 1775	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
16	9, 15	nfxfrd 1489	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
17	8, 16	sylbir 135	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
18	6, 17	jaoi 717	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
19	5, 18	jaoi 717	. 2 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
20	1, 19	ax-mp 5	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1362  Ⅎwnf 1474  ∃wex 1506  [wsb 1776

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  nfsb  1965  sbalyz  2018  opelopabsb  4294  bezoutlemmain  12165