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Theorem hbsb 2429
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.)
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)
21nf5i 2011 . . 3 𝑧𝜑
32nfsb 2428 . 2 𝑧[𝑦 / 𝑥]𝜑
43nf5ri 2053 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  [wsb 1867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868
This theorem is referenced by:  hbab  2601  hblem  2718  bj-hblem  31837
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