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Theorem nfsbxyt 1867
Description: Closed form of nfsbxy 1866. (Contributed by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsbxyt (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbxyt
StepHypRef Expression
1 ax-bndl 1444 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 nfs1v 1863 . . . . 5 𝑧[𝑦 / 𝑧]𝜑
3 drsb1 1727 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
43drnf2 1669 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
52, 4mpbii 146 . . . 4 (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
65a1d 22 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
7 a16nf 1794 . . . . 5 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
87a1d 22 . . . 4 (∀𝑧 𝑧 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
9 df-nf 1395 . . . . . 6 (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
109albii 1404 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 ↔ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11 sb5 1815 . . . . . . 7 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
12 nfa1 1479 . . . . . . . . 9 𝑥𝑥𝑧 𝑥 = 𝑦
13 nfa1 1479 . . . . . . . . 9 𝑥𝑥𝑧𝜑
1412, 13nfan 1502 . . . . . . . 8 𝑥(∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑)
15 sp 1446 . . . . . . . . . 10 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
1615adantr 270 . . . . . . . . 9 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧 𝑥 = 𝑦)
17 sp 1446 . . . . . . . . . 10 (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜑)
1817adantl 271 . . . . . . . . 9 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧𝜑)
1916, 18nfand 1505 . . . . . . . 8 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧(𝑥 = 𝑦𝜑))
2014, 19nfexd 1691 . . . . . . 7 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧𝑥(𝑥 = 𝑦𝜑))
2111, 20nfxfrd 1409 . . . . . 6 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
2221ex 113 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
2310, 22sylbir 133 . . . 4 (∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
248, 23jaoi 671 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
256, 24jaoi 671 . 2 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
261, 25ax-mp 7 1 (∀𝑥𝑧𝜑 → Ⅎ𝑧[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:  nfsbt  1898
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