Theorem nfsbv 2349

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 2565 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 2342	. . . 4 ⊢ Ⅎ𝑧∀𝑥(𝑥 = 𝑤 → 𝜑)
7	2, 6	nfim 1897	. . 3 ⊢ Ⅎ𝑧(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))
8	7	nfal 2342	. 2 ⊢ Ⅎ𝑧∀𝑤(𝑤 = 𝑦 → ∀𝑥(𝑥 = 𝑤 → 𝜑))
9	1, 8	nfxfr 1853	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4  ∀wal 1535  Ⅎwnf 1784  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by:  hbsbw  2351  sbco2v  2352  2sb8ev  2375  sb8euv  2685  2mo  2733  nfcrii  2970  cbvralfw  3437  cbvreuw  3443  cbvrabw  3489  cbvrabcsfw  3924  cbvopab1  5139  cbvmptf  5165  ralxpf  5717  cbviotaw  6321  cbvriotaw  7123  dfoprab4f  7754  mo5f  30253  ax11-pm2  34159  dfich2  43662  dfich2ai  43663  ichbi12i  43667