Theorem nfsbv 2329

Description: If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is disjoint from both 𝑥 and 𝑦. Version of nfsb 2526 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2375. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)

Hypothesis

Ref	Expression
nfsbv.nf	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsbv	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem nfsbv

Step	Hyp	Ref	Expression
1		nfsbv.nf	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	nf5ri 2193	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
3	2	hbsbw 2169	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
4	3	nf5i 2144	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1780  [wsb 2062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-ex 1777  df-nf 1781  df-sb 2063

This theorem is referenced by:  sbco2v  2331  2sb8ef  2357  sb8euv  2597  2mo  2646  cbvreuwOLD  3413  cbvrabwOLD  3472  cbvrabcsfw  3952  cbvopab1  5223  cbvmptf  5257  ralxpf  5860  cbviotaw  6523  cbvriotaw  7397  dfoprab4f  8080  mo5f  32517  ax11-pm2  36819  dfich2  47383  ichbi12i  47385