Theorem nfsb 2561

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 2369 for a version with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2410. (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 2410. Use nfsbv 2369 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 1831	. . 3 ⊢ Ⅎ𝑥⊤
2		nfsb.1	. . . 4 ⊢ Ⅎ𝑧𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
4	1, 3	nfsbd 2560	. 2 ⊢ (⊤ → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
5	4	mptru 1574	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  ⊤wtru 1568  Ⅎwnf 1810  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098

This theorem is referenced by:  hbsb  2562  sb10f  2565  2sb8e  2568  sb8eu  2634  cbvralf  3356  cbvralsv  3362  cbvrexsv  3363  cbvreu  3415  cbvrab  3462  cbvreucsf  3905  cbvrabcsf  3906  cbvopab1g  5187  cbvmptfg  5213  cbviota  6498  sb8iota  6500  cbvriota  7378  2sb5nd  45154