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Theorem nfsb 1917
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)
Hypothesis
Ref Expression
nfsb.1 𝑧𝜑
Assertion
Ref Expression
nfsb 𝑧[𝑦 / 𝑥]𝜑
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfsb.1 . . . 4 𝑧𝜑
21nfsbxy 1913 . . 3 𝑧[𝑤 / 𝑥]𝜑
32nfsbxy 1913 . 2 𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4 ax-17 1506 . . . 4 (𝜑 → ∀𝑤𝜑)
54sbco2vh 1916 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
65nfbii 1449 . 2 (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
73, 6mpbi 144 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  wnf 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:  nfsbv  1918  hbsb  1920  sbco2yz  1934  sbcomxyyz  1943  hbsbd  1955  nfsb4or  1996  sb8eu  2010  nfeu  2016  cbvab  2261  cbvralf  2646  cbvrexf  2647  cbvreu  2650  cbvralsv  2663  cbvrexsv  2664  cbvrab  2679  cbvreucsf  3059  cbvrabcsf  3060  cbvopab1  3996  cbvmptf  4017  cbvmpt  4018  ralxpf  4680  rexxpf  4681  cbviota  5088  sb8iota  5090  cbvriota  5733  dfoprab4f  6084
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