Theorem nfsb 2557

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 2365 for a version with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2406. (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 2406. Use nfsbv 2365 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
nfsb.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsb	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable group:   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfsb

Step	Hyp	Ref	Expression
1		nftru 1827	. . 3 ⊢ Ⅎ𝑥⊤
2		nfsb.1	. . . 4 ⊢ Ⅎ𝑧𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
4	1, 3	nfsbd 2556	. 2 ⊢ (⊤ → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
5	4	mptru 1570	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  ⊤wtru 1564  Ⅎwnf 1806  [wsb 2093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094

This theorem is referenced by:  hbsb  2558  sb10f  2561  2sb8e  2564  sb8eu  2630  cbvralf  3350  cbvralsv  3356  cbvrexsv  3357  cbvreu  3409  cbvrab  3456  cbvreucsf  3899  cbvrabcsf  3900  cbvopab1g  5180  cbvmptfg  5206  cbviota  6490  sb8iota  6492  cbvriota  7370  2sb5nd  45134