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Theorem hbsb 1871
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
Hypothesis
Ref Expression
hbsb.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
hbsb ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbsb
StepHypRef Expression
1 hbsb.1 . . . 4 (𝜑 → ∀𝑧𝜑)
21nfi 1396 . . 3 𝑧𝜑
32nfsb 1870 . 2 𝑧[𝑦 / 𝑥]𝜑
43nfri 1457 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  [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:  sb10f  1919  hbsb4  1936  sb8euh  1971  hbab  2079  hblem  2195
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