Theorem sbcomxyyz 1972

Description: Version of sbcom 1975 with distinct variable constraints between 𝑥 and 𝑦, and 𝑦 and 𝑧. (Contributed by Jim Kingdon, 21-Mar-2018.)

Assertion

Ref	Expression
sbcomxyyz	⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝑧

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

Proof of Theorem sbcomxyyz

Step	Hyp	Ref	Expression
1		ax-bndl 1509	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		ax-ial 1534	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ∀𝑧∀𝑧 𝑧 = 𝑥)
3		drsb1 1799	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	sbbidh 1845	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
5		drsb1 1799	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
6	4, 5	bitr3d 190	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
7		sbequ12 1771	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
8	7	sps 1537	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
9		hbae 1718	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑥∀𝑧 𝑧 = 𝑦)
10		sbequ12 1771	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
11	10	sps 1537	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
12	9, 11	sbbidh 1845	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
13	8, 12	bitr3d 190	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
14		df-nf 1461	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
15	14	albii 1470	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
16		ax-ial 1534	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ∀𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦)
17		nfs1v 1939	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
18	17	nfsb 1946	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
19	18	a1i 9	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
20	19	nfrd 1520	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
21		nfr 1518	. . . . . . . . 9 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
22		nfnf1 1544	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧Ⅎ𝑧 𝑥 = 𝑦
23		nfa1 1541	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∀𝑧 𝑥 = 𝑦
24	22, 23	nfan 1565	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦)
25	24	nfri 1519	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ∀𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦))
26		nfs1v 1939	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
27	26	a1i 9	. . . . . . . . . . . 12 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
28	27	nfrd 1520	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
29		sbequ12 1771	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
30	29, 7	sylan9bb 462	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
31	30	ex 115	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
32	31	sps 1537	. . . . . . . . . . . 12 ⊢ (∀𝑧 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
33	32	adantl 277	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
34	25, 28, 33	sbiedh 1787	. . . . . . . . . 10 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
35	34	ex 115	. . . . . . . . 9 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
36	21, 35	syld 45	. . . . . . . 8 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
37	36	sps 1537	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
38	16, 20, 37	sbiedh 1787	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
39	38	bicomd 141	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
40	15, 39	sylbir 135	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
41	13, 40	jaoi 716	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
42	6, 41	jaoi 716	. 2 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
43	1, 42	ax-mp 5	1 ⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  ∀wal 1351  Ⅎwnf 1460  [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbco3xzyz  1973