Theorem sbcomxyyz 1965

Description: Version of sbcom 1968 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 1502	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		ax-ial 1527	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ∀𝑧∀𝑧 𝑧 = 𝑥)
3		drsb1 1792	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	sbbidh 1838	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
5		drsb1 1792	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
6	4, 5	bitr3d 189	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
7		sbequ12 1764	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
8	7	sps 1530	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
9		hbae 1711	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → ∀𝑥∀𝑧 𝑧 = 𝑦)
10		sbequ12 1764	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
11	10	sps 1530	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
12	9, 11	sbbidh 1838	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
13	8, 12	bitr3d 189	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
14		df-nf 1454	. . . . . 6 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
15	14	albii 1463	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
16		ax-ial 1527	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ∀𝑥∀𝑥Ⅎ𝑧 𝑥 = 𝑦)
17		nfs1v 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
18	17	nfsb 1939	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
19	18	a1i 9	. . . . . . . 8 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
20	19	nfrd 1513	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
21		nfr 1511	. . . . . . . . 9 ⊢ (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
22		nfnf1 1537	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧Ⅎ𝑧 𝑥 = 𝑦
23		nfa1 1534	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧∀𝑧 𝑥 = 𝑦
24	22, 23	nfan 1558	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦)
25	24	nfri 1512	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ∀𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦))
26		nfs1v 1932	. . . . . . . . . . . . 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 1513	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
29		sbequ12 1764	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
30	29, 7	sylan9bb 459	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑦) → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
31	30	ex 114	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
32	31	sps 1530	. . . . . . . . . . . 12 ⊢ (∀𝑧 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
33	32	adantl 275	. . . . . . . . . . 11 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
34	25, 28, 33	sbiedh 1780	. . . . . . . . . 10 ⊢ ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
35	34	ex 114	. . . . . . . . 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 1530	. . . . . . 7 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
38	16, 20, 37	sbiedh 1780	. . . . . 6 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
39	38	bicomd 140	. . . . 5 ⊢ (∀𝑥Ⅎ𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
40	15, 39	sylbir 134	. . . 4 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
41	13, 40	jaoi 711	. . 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 711	. 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 103   ↔ wb 104   ∨ wo 703  ∀wal 1346  Ⅎwnf 1453  [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  ax-i5r 1528

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

This theorem is referenced by:  sbco3xzyz  1966