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Theorem nfsbxyt 1862
Description: Closed form of nfsbxy 1861. (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 1440 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 nfs1v 1858 . . . . 5 𝑧[𝑦 / 𝑧]𝜑
3 drsb1 1722 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
43drnf2 1664 . . . . 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 1789 . . . . 5 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
87a1d 22 . . . 4 (∀𝑧 𝑧 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
9 df-nf 1391 . . . . . 6 (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
109albii 1400 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 ↔ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11 sb5 1810 . . . . . . 7 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
12 nfa1 1475 . . . . . . . . 9 𝑥𝑥𝑧 𝑥 = 𝑦
13 nfa1 1475 . . . . . . . . 9 𝑥𝑥𝑧𝜑
1412, 13nfan 1498 . . . . . . . 8 𝑥(∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑)
15 sp 1442 . . . . . . . . . 10 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
1615adantr 270 . . . . . . . . 9 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧 𝑥 = 𝑦)
17 sp 1442 . . . . . . . . . 10 (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜑)
1817adantl 271 . . . . . . . . 9 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧𝜑)
1916, 18nfand 1501 . . . . . . . 8 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧(𝑥 = 𝑦𝜑))
2014, 19nfexd 1686 . . . . . . 7 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧𝑥(𝑥 = 𝑦𝜑))
2111, 20nfxfrd 1405 . . . . . 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 669 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
256, 24jaoi 669 . 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 662  wal 1283  wnf 1390  wex 1422  [wsb 1687
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  nfsbt  1893
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