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Theorem nfsbt 1969
Description: Closed form of nfsb 1939. (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 1519 . 2 (∀𝑥𝑧𝜑 → ∀𝑤𝑥𝑧𝜑)
2 nfsbxyt 1936 . . . . 5 (∀𝑥𝑧𝜑 → Ⅎ𝑧[𝑤 / 𝑥]𝜑)
32alimi 1448 . . . 4 (∀𝑤𝑥𝑧𝜑 → ∀𝑤𝑧[𝑤 / 𝑥]𝜑)
4 nfsbxyt 1936 . . . 4 (∀𝑤𝑧[𝑤 / 𝑥]𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
53, 4syl 14 . . 3 (∀𝑤𝑥𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑)
6 nfv 1521 . . . . 5 𝑤𝜑
76sbco2 1958 . . . 4 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
87nfbii 1466 . . 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 1346  wnf 1453  [wsb 1755
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756
This theorem is referenced by:  nfsbd  1970  setindft  14000
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