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Theorem nfsb 2525
Description: If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. See nfsbv 2328 for a version with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Feb-2024.) Usage of this theorem is discouraged because it depends on ax-13 2374. Use nfsbv 2328 instead. (New usage is discouraged.)
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
StepHypRef Expression
1 nftru 1800 . . 3 𝑥
2 nfsb.1 . . . 4 𝑧𝜑
32a1i 11 . . 3 (⊤ → Ⅎ𝑧𝜑)
41, 3nfsbd 2524 . 2 (⊤ → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
54mptru 1543 1 𝑧[𝑦 / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wtru 1537  wnf 1779  [wsb 2061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-10 2138  ax-11 2154  ax-12 2174  ax-13 2374
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062
This theorem is referenced by:  hbsb  2526  sb10f  2529  2sb8e  2532  sb8eu  2597  cbvralf  3357  cbvralsv  3363  cbvrexsv  3364  cbvreu  3424  cbvrab  3476  cbvreucsf  3954  cbvrabcsf  3955  cbvopab1g  5223  cbvmptfg  5257  cbviota  6524  sb8iota  6526  cbvriota  7400  2sb5nd  44557
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