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Theorem sbcomxyyz 1921
Description: Version of sbcom 1924 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 1469 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 ax-ial 1497 . . . . 5 (∀𝑧 𝑧 = 𝑥 → ∀𝑧𝑧 𝑧 = 𝑥)
3 drsb1 1753 . . . . 5 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3sbbidh 1799 . . . 4 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
5 drsb1 1753 . . . 4 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
64, 5bitr3d 189 . . 3 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
7 sbequ12 1727 . . . . . 6 (𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
87sps 1500 . . . . 5 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
9 hbae 1679 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → ∀𝑥𝑧 𝑧 = 𝑦)
10 sbequ12 1727 . . . . . . 7 (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
1110sps 1500 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧]𝜑))
129, 11sbbidh 1799 . . . . 5 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
138, 12bitr3d 189 . . . 4 (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
14 df-nf 1420 . . . . . 6 (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
1514albii 1429 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 ↔ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
16 ax-ial 1497 . . . . . . 7 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑥𝑥𝑧 𝑥 = 𝑦)
17 nfs1v 1890 . . . . . . . . . 10 𝑥[𝑦 / 𝑥]𝜑
1817nfsb 1897 . . . . . . . . 9 𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
1918a1i 9 . . . . . . . 8 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
2019nfrd 1483 . . . . . . 7 (∀𝑥𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
21 nfr 1481 . . . . . . . . 9 (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
22 nfnf1 1506 . . . . . . . . . . . . 13 𝑧𝑧 𝑥 = 𝑦
23 nfa1 1504 . . . . . . . . . . . . 13 𝑧𝑧 𝑥 = 𝑦
2422, 23nfan 1527 . . . . . . . . . . . 12 𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦)
2524nfri 1482 . . . . . . . . . . 11 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ∀𝑧(Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦))
26 nfs1v 1890 . . . . . . . . . . . . 13 𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑
2726a1i 9 . . . . . . . . . . . 12 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑)
2827nfrd 1483 . . . . . . . . . . 11 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
29 sbequ12 1727 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
3029, 7sylan9bb 455 . . . . . . . . . . . . . 14 ((𝑥 = 𝑦𝑧 = 𝑦) → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
3130ex 114 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3231sps 1500 . . . . . . . . . . . 12 (∀𝑧 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3332adantl 273 . . . . . . . . . . 11 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → (𝑧 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3425, 28, 33sbiedh 1743 . . . . . . . . . 10 ((Ⅎ𝑧 𝑥 = 𝑦 ∧ ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
3534ex 114 . . . . . . . . 9 (Ⅎ𝑧 𝑥 = 𝑦 → (∀𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3621, 35syld 45 . . . . . . . 8 (Ⅎ𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3736sps 1500 . . . . . . 7 (∀𝑥𝑧 𝑥 = 𝑦 → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑)))
3816, 20, 37sbiedh 1743 . . . . . 6 (∀𝑥𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑))
3938bicomd 140 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
4015, 39sylbir 134 . . . 4 (∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
4113, 40jaoi 688 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑))
426, 41jaoi 688 . 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 5 1 ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 680  wal 1312  wnf 1419  [wsb 1718
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  sbco3xzyz  1922
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