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Theorem nfsbv 2362
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑥, 𝑦 and 𝑧 are distinct. Version of nfsb 2575 requiring more disjoint variables, but fewer axioms. (Contributed by Wolf Lammen, 7-Feb-2023.)
Hypothesis
Ref Expression
nfsbv.nf 𝑧𝜑
Assertion
Ref Expression
nfsbv 𝑧[𝑦 / 𝑥]𝜑
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsbv
StepHypRef Expression
1 sb6 2307 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2 nfv 2013 . . . 4 𝑧 𝑥 = 𝑦
3 nfsbv.nf . . . 4 𝑧𝜑
42, 3nfim 1999 . . 3 𝑧(𝑥 = 𝑦𝜑)
54nfal 2355 . 2 𝑧𝑥(𝑥 = 𝑦𝜑)
61, 5nfxfr 1952 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1654  wnf 1882  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068
This theorem is referenced by:  2sb8ev  2378  sb8euv  2687
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