Theorem nfsbv 2338

Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is distinct from 𝑥 and 𝑦. Version of nfsb 2542 requiring more disjoint variables, but fewer axioms. (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.)

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

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sb 2070	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑)))
2		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝑦
3		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧 𝑥 = 𝑤
4		nfsbv.nf	. . . . . 6 ⊢ Ⅎ𝑧𝜑
5	3, 4	nfim 1897	. . . . 5 ⊢ Ⅎ𝑧(𝑥 = 𝑤 → 𝜑)
6	5	nfal 2331	. . . 4 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑤 → 𝜑)
7	2, 6	nfim 1897	. . 3 ⊢ Ⅎ𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))
8	7	nfal 2331	. 2 ⊢ Ⅎ𝑧∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))
9	1, 8	nfxfr 1854	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070

This theorem is referenced by:  hbsbwOLD  2340  sbco2v  2341  2sb8ev  2364  sb8euv  2660  2mo  2710  nfcriiOLD  2949  cbvralfwOLD  3383  cbvreuw  3389  cbvrabw  3437  cbvrabcsfw  3869  cbvopab1  5103  cbvmptf  5129  ralxpf  5681  cbviotaw  6290  cbvriotaw  7102  dfoprab4f  7736  mo5f  30260  ax11-pm2  34274  dfich2  43975  ichbi12i  43977