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Theorem nfsbt 1947
 Description: Closed form of nfsb 1917. (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 1506 . 2 (∀𝑥𝑧𝜑 → ∀𝑤𝑥𝑧𝜑)
2 nfsbxyt 1914 . . . . 5 (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑤 / 𝑥]𝜑)
32alimi 1431 . . . 4 (∀𝑤𝑥𝑧𝜑 → ∀𝑤𝑧[𝑤 / 𝑥]𝜑)
4 nfsbxyt 1914 . . . 4 (∀𝑤𝑧[𝑤 / 𝑥]𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
53, 4syl 14 . . 3 (∀𝑤𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
6 nfv 1508 . . . . 5 𝑤𝜑
76sbco2 1936 . . . 4 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
87nfbii 1449 . . 3 (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
95, 8sylib 121 . 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 1329  Ⅎwnf 1436  [wsb 1735 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736 This theorem is referenced by:  nfsbd  1948  setindft  13152
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