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Theorem nfsb 1838
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 1834 . . 3 𝑧[𝑤 / 𝑥]𝜑
32nfsbxy 1834 . 2 𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4 ax-17 1435 . . . 4 (𝜑 → ∀𝑤𝜑)
54sbco2v 1837 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
65nfbii 1378 . 2 (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
73, 6mpbi 137 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  wnf 1365  [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  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  hbsb  1839  sbco2yz  1853  sbcomxyyz  1862  hbsbd  1874  nfsb4or  1915  sb8eu  1929  nfeu  1935  cbvab  2176  cbvralf  2544  cbvrexf  2545  cbvreu  2548  cbvralsv  2561  cbvrexsv  2562  cbvrab  2572  cbvreucsf  2938  cbvrabcsf  2939  cbvopab1  3858  cbvmpt  3879  ralxpf  4510  rexxpf  4511  cbviota  4900  sb8iota  4902  cbvriota  5506  dfoprab4f  5847
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