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Theorem nfsbxyt 1835
Description: Closed form of nfsbxy 1834. (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 1415 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 nfs1v 1831 . . . . 5 𝑧[𝑦 / 𝑧]𝜑
3 drsb1 1696 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑))
43drnf2 1638 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (Ⅎ𝑧[𝑦 / 𝑧]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑))
52, 4mpbii 140 . . . 4 (∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
65a1d 22 . . 3 (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
7 a16nf 1762 . . . . 5 (∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
87a1d 22 . . . 4 (∀𝑧 𝑧 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
9 df-nf 1366 . . . . . 6 (Ⅎ𝑧 𝑥 = 𝑦 ↔ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
109albii 1375 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 ↔ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11 sb5 1783 . . . . . . 7 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
12 nfa1 1450 . . . . . . . . 9 𝑥𝑥𝑧 𝑥 = 𝑦
13 nfa1 1450 . . . . . . . . 9 𝑥𝑥𝑧𝜑
1412, 13nfan 1473 . . . . . . . 8 𝑥(∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑)
15 sp 1417 . . . . . . . . . 10 (∀𝑥𝑧 𝑥 = 𝑦 → Ⅎ𝑧 𝑥 = 𝑦)
1615adantr 265 . . . . . . . . 9 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧 𝑥 = 𝑦)
17 sp 1417 . . . . . . . . . 10 (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜑)
1817adantl 266 . . . . . . . . 9 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧𝜑)
1916, 18nfand 1476 . . . . . . . 8 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧(𝑥 = 𝑦𝜑))
2014, 19nfexd 1660 . . . . . . 7 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧𝑥(𝑥 = 𝑦𝜑))
2111, 20nfxfrd 1380 . . . . . 6 ((∀𝑥𝑧 𝑥 = 𝑦 ∧ ∀𝑥𝑧𝜑) → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
2221ex 112 . . . . 5 (∀𝑥𝑧 𝑥 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
2310, 22sylbir 129 . . . 4 (∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
248, 23jaoi 646 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑))
256, 24jaoi 646 . 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 101  wo 639  wal 1257  wnf 1365  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  nfsbt  1866
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