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Theorem sbcomxyyz 1843
Description: Version of sbcom 1846 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-bnd 1396 . 2 (z z = x (z z = y xz(x = yz x = y)))
2 ax-ial 1424 . . . . 5 (z z = xzz z = x)
3 drsb1 1677 . . . . 5 (z z = x → ([y / z]φ ↔ [y / x]φ))
42, 3sbbid 1723 . . . 4 (z z = x → ([y / z][y / z]φ ↔ [y / z][y / x]φ))
5 drsb1 1677 . . . 4 (z z = x → ([y / z][y / z]φ ↔ [y / x][y / z]φ))
64, 5bitr3d 179 . . 3 (z z = x → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
7 sbequ12 1651 . . . . . 6 (z = y → ([y / x]φ ↔ [y / z][y / x]φ))
87sps 1427 . . . . 5 (z z = y → ([y / x]φ ↔ [y / z][y / x]φ))
9 hbae 1603 . . . . . 6 (z z = yxz z = y)
10 sbequ12 1651 . . . . . . 7 (z = y → (φ ↔ [y / z]φ))
1110sps 1427 . . . . . 6 (z z = y → (φ ↔ [y / z]φ))
129, 11sbbid 1723 . . . . 5 (z z = y → ([y / x]φ ↔ [y / x][y / z]φ))
138, 12bitr3d 179 . . . 4 (z z = y → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
14 df-nf 1347 . . . . . 6 (Ⅎz x = yz(x = yz x = y))
1514albii 1356 . . . . 5 (xz x = yxz(x = yz x = y))
16 ax-ial 1424 . . . . . . 7 (xz x = yxxz x = y)
17 nfs1v 1812 . . . . . . . . . 10 x[y / x]φ
1817nfsb 1819 . . . . . . . . 9 x[y / z][y / x]φ
1918a1i 9 . . . . . . . 8 (xz x = y → Ⅎx[y / z][y / x]φ)
2019nfrd 1410 . . . . . . 7 (xz x = y → ([y / z][y / x]φx[y / z][y / x]φ))
21 nfr 1408 . . . . . . . . 9 (Ⅎz x = y → (x = yz x = y))
22 nfnf1 1433 . . . . . . . . . . . . 13 zz x = y
23 nfa1 1431 . . . . . . . . . . . . 13 zz x = y
2422, 23nfan 1454 . . . . . . . . . . . 12 z(Ⅎz x = y z x = y)
2524nfri 1409 . . . . . . . . . . 11 ((Ⅎz x = y z x = y) → z(Ⅎz x = y z x = y))
26 nfs1v 1812 . . . . . . . . . . . . 13 z[y / z][y / x]φ
2726a1i 9 . . . . . . . . . . . 12 ((Ⅎz x = y z x = y) → Ⅎz[y / z][y / x]φ)
2827nfrd 1410 . . . . . . . . . . 11 ((Ⅎz x = y z x = y) → ([y / z][y / x]φz[y / z][y / x]φ))
29 sbequ12 1651 . . . . . . . . . . . . . . 15 (x = y → (φ ↔ [y / x]φ))
3029, 7sylan9bb 435 . . . . . . . . . . . . . 14 ((x = y z = y) → (φ ↔ [y / z][y / x]φ))
3130ex 108 . . . . . . . . . . . . 13 (x = y → (z = y → (φ ↔ [y / z][y / x]φ)))
3231sps 1427 . . . . . . . . . . . 12 (z x = y → (z = y → (φ ↔ [y / z][y / x]φ)))
3332adantl 262 . . . . . . . . . . 11 ((Ⅎz x = y z x = y) → (z = y → (φ ↔ [y / z][y / x]φ)))
3425, 28, 33sbiedh 1667 . . . . . . . . . 10 ((Ⅎz x = y z x = y) → ([y / z]φ ↔ [y / z][y / x]φ))
3534ex 108 . . . . . . . . 9 (Ⅎz x = y → (z x = y → ([y / z]φ ↔ [y / z][y / x]φ)))
3621, 35syld 40 . . . . . . . 8 (Ⅎz x = y → (x = y → ([y / z]φ ↔ [y / z][y / x]φ)))
3736sps 1427 . . . . . . 7 (xz x = y → (x = y → ([y / z]φ ↔ [y / z][y / x]φ)))
3816, 20, 37sbiedh 1667 . . . . . 6 (xz x = y → ([y / x][y / z]φ ↔ [y / z][y / x]φ))
3938bicomd 129 . . . . 5 (xz x = y → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
4015, 39sylbir 125 . . . 4 (xz(x = yz x = y) → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
4113, 40jaoi 635 . . 3 ((z z = y xz(x = yz x = y)) → ([y / z][y / x]φ ↔ [y / x][y / z]φ))
426, 41jaoi 635 . 2 ((z z = x (z z = y xz(x = yz 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 97  wb 98   wo 628  wal 1240  wnf 1346  [wsb 1642
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sbco3xzyz  1844
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