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Theorem hbsb 1823
 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 1351 . . 3 𝑧𝜑
32nfsb 1822 . 2 𝑧[𝑦 / 𝑥]𝜑
43nfri 1412 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  sb10f  1871  hbsb4  1888  sb8euh  1923  hbab  2031  hblem  2145
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