Theorem sbcomxyyz 1846

Description: Version of sbcom 1849 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 1399	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		ax-ial 1427	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ∀𝑧∀𝑧 𝑧 = 𝑥)
3		drsb1 1680	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	sbbid 1726	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
5		drsb1 1680	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
6	4, 5	bitr3d 179	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
7		sbequ12 1654	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
8	7	sps 1430	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
9		hbae 1606	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑥∀𝑧 𝑧 = 𝑦)
10		sbequ12 1654	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
11	10	sps 1430	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
12	9, 11	sbbid 1726	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
13	8, 12	bitr3d 179	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
14		df-nf 1350	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
15	14	albii 1359	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
16		ax-ial 1427	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ∀𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦)
17		nfs1v 1815	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
18	17	nfsb 1822	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
19	18	a1i 9	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
20	19	nfrd 1413	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
21		nfr 1411	. . . . . . . . 9 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
22		nfnf1 1436	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧Ⅎ𝑧 𝑥 = 𝑦
23		nfa1 1434	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∀𝑧 𝑥 = 𝑦
24	22, 23	nfan 1457	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦)
25	24	nfri 1412	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ∀𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦))
26		nfs1v 1815	. . . . . . . . . . . . 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 1413	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
29		sbequ12 1654	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
30	29, 7	sylan9bb 435	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
31	30	ex 108	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
32	31	sps 1430	. . . . . . . . . . . 12 ⊢ (∀𝑧 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
33	32	adantl 262	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
34	25, 28, 33	sbiedh 1670	. . . . . . . . . 10 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
35	34	ex 108	. . . . . . . . 9 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
36	21, 35	syld 40	. . . . . . . 8 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
37	36	sps 1430	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
38	16, 20, 37	sbiedh 1670	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
39	38	bicomd 129	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
40	15, 39	sylbir 125	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
41	13, 40	jaoi 636	. . 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 636	. 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 7	1 ⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  ∀wal 1241  Ⅎwnf 1349  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbco3xzyz  1847