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Theorem nfsbt 1892
Description: Closed form of nfsb 1864. (Contributed by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsbt (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbt
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-17 1460 . 2 (∀𝑥𝑧𝜑 → ∀𝑤𝑥𝑧𝜑)
2 nfsbxyt 1861 . . . . 5 (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑤 / 𝑥]𝜑)
32alimi 1385 . . . 4 (∀𝑤𝑥𝑧𝜑 → ∀𝑤𝑧[𝑤 / 𝑥]𝜑)
4 nfsbxyt 1861 . . . 4 (∀𝑤𝑧[𝑤 / 𝑥]𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
53, 4syl 14 . . 3 (∀𝑤𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
6 nfv 1462 . . . . 5 𝑤𝜑
76sbco2 1881 . . . 4 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
87nfbii 1403 . . 3 (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
95, 8sylib 120 . 2 (∀𝑤𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
101, 9syl 14 1 (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390  [wsb 1686
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  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  nfsbd  1893  setindft  10945
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