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Theorem sbcomxyyz 1862
 Description: Version of sbcom 1865 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
StepHypRef Expression
1 ax-bndl 1415 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 ax-ial 1443 . . . . 5 (∀𝑧 𝑧 = 𝑥 → ∀𝑧𝑧 𝑧 = 𝑥)
3 drsb1 1696 . . . . 5 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3sbbid 1742 . . . 4 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
5 drsb1 1696 . . . 4 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
64, 5bitr3d 183 . . 3 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
7 sbequ12 1670 . . . . . 6 (𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
87sps 1446 . . . . 5 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
9 hbae 1622 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → ∀𝑥𝑧 𝑧 = 𝑦)
10 sbequ12 1670 . . . . . . 7 (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
1110sps 1446 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
129, 11sbbid 1742 . . . . 5 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
138, 12bitr3d 183 . . . 4 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
14 df-nf 1366 . . . . . 6 (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
1514albii 1375 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 ↔ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
16 ax-ial 1443 . . . . . . 7 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑥𝑥𝑧 𝑥 = 𝑦)
17 nfs1v 1831 . . . . . . . . . 10 𝑥[𝑦 / 𝑥]𝜑
1817nfsb 1838 . . . . . . . . 9 𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
1918a1i 9 . . . . . . . 8 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
2019nfrd 1429 . . . . . . 7 (∀𝑥𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
21 nfr 1427 . . . . . . . . 9 (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
22 nfnf1 1452 . . . . . . . . . . . . 13 𝑧𝑧 𝑥 = 𝑦
23 nfa1 1450 . . . . . . . . . . . . 13 𝑧𝑧 𝑥 = 𝑦
2422, 23nfan 1473 . . . . . . . . . . . 12 𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦)
2524nfri 1428 . . . . . . . . . . 11 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ∀𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦))
26 nfs1v 1831 . . . . . . . . . . . . 13 𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
2726a1i 9 . . . . . . . . . . . 12 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
2827nfrd 1429 . . . . . . . . . . 11 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
29 sbequ12 1670 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
3029, 7sylan9bb 443 . . . . . . . . . . . . . 14 ((𝑥 = 𝑦𝑧 = 𝑦) → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
3130ex 112 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3231sps 1446 . . . . . . . . . . . 12 (∀𝑧 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3332adantl 266 . . . . . . . . . . 11 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3425, 28, 33sbiedh 1686 . . . . . . . . . 10 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
3534ex 112 . . . . . . . . 9 (Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3621, 35syld 44 . . . . . . . 8 (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3736sps 1446 . . . . . . 7 (∀𝑥𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3816, 20, 37sbiedh 1686 . . . . . 6 (∀𝑥𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
3938bicomd 133 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
4015, 39sylbir 129 . . . 4 (∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
4113, 40jaoi 646 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
426, 41jaoi 646 . 2 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
431, 42ax-mp 7 1 ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   ∨ wo 639  ∀wal 1257  Ⅎwnf 1365  [wsb 1661 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662 This theorem is referenced by:  sbco3xzyz  1863
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