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Theorem nfsb 1870
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 1866 . . 3 𝑧[𝑤 / 𝑥]𝜑
32nfsbxy 1866 . 2 𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4 ax-17 1464 . . . 4 (𝜑 → ∀𝑤𝜑)
54sbco2v 1869 . . 3 ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
65nfbii 1407 . 2 (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
73, 6mpbi 143 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff set class
Syntax hints:  wnf 1394  [wsb 1692
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  hbsb  1871  sbco2yz  1885  sbcomxyyz  1894  hbsbd  1906  nfsb4or  1947  sb8eu  1961  nfeu  1967  cbvab  2210  cbvralf  2584  cbvrexf  2585  cbvreu  2588  cbvralsv  2601  cbvrexsv  2602  cbvrab  2617  cbvreucsf  2992  cbvrabcsf  2993  cbvopab1  3911  cbvmptf  3932  cbvmpt  3933  ralxpf  4582  rexxpf  4583  cbviota  4985  sb8iota  4987  cbvriota  5618  dfoprab4f  5963
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