Theorem nfsbv 2000

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 1999 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by GG, 25-Aug-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		df-sb 1811	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		nfv 1576	. . . 4 ⊢ Ⅎ𝑧 𝑥 = 𝑦
3		nfsbv.nf	. . . 4 ⊢ Ⅎ𝑧𝜑
4	2, 3	nfim 1620	. . 3 ⊢ Ⅎ𝑧(𝑥 = 𝑦 → 𝜑)
5	2, 3	nfan 1613	. . . 4 ⊢ Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑)
6	5	nfex 1685	. . 3 ⊢ Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
7	4, 6	nfan 1613	. 2 ⊢ Ⅎ𝑧((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8	1, 7	nfxfr 1522	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1508  ∃wex 1540  [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sbco2v  2001  cbvabw  2354