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Theorem nfsbxy 1866
Description: Similar to hbsb 1871 but with an extra distinct variable constraint, on 𝑥 and 𝑦. (Contributed by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
nfsbxy.1 𝑧𝜑
Assertion
Ref Expression
nfsbxy 𝑧[𝑦 / 𝑥]𝜑
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbxy
StepHypRef Expression
1 ax-bndl 1444 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 nfs1v 1863 . . . 4 𝑧[𝑦 / 𝑧]𝜑
3 drsb1 1727 . . . . 5 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
43drnf2 1669 . . . 4 (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
52, 4mpbii 146 . . 3 (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
6 a16nf 1794 . . . 4 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7 df-nf 1395 . . . . . 6 (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
87albii 1404 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 ↔ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9 sb5 1815 . . . . . 6 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
10 nfa1 1479 . . . . . . 7 𝑥𝑥𝑧 𝑥 = 𝑦
11 sp 1446 . . . . . . . 8 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
12 nfsbxy.1 . . . . . . . . 9 𝑧𝜑
1312a1i 9 . . . . . . . 8 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
1411, 13nfand 1505 . . . . . . 7 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧(𝑥 = 𝑦𝜑))
1510, 14nfexd 1691 . . . . . 6 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧𝑥(𝑥 = 𝑦𝜑))
169, 15nfxfrd 1409 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
178, 16sylbir 133 . . . 4 (∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
186, 17jaoi 671 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
195, 18jaoi 671 . 2 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
201, 19ax-mp 7 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 664  wal 1287  wnf 1394  wex 1426  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  nfsb  1870  sbalyz  1923  opelopabsb  4087  bezoutlemmain  11265
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