Theorem nfsbv 1975

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 1974 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 1786	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		nfv 1551	. . . 4 ⊢ Ⅎ𝑧 𝑥 = 𝑦
3		nfsbv.nf	. . . 4 ⊢ Ⅎ𝑧𝜑
4	2, 3	nfim 1595	. . 3 ⊢ Ⅎ𝑧(𝑥 = 𝑦 → 𝜑)
5	2, 3	nfan 1588	. . . 4 ⊢ Ⅎ𝑧(𝑥 = 𝑦 ∧ 𝜑)
6	5	nfex 1660	. . 3 ⊢ Ⅎ𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
7	4, 6	nfan 1588	. 2 ⊢ Ⅎ𝑧((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8	1, 7	nfxfr 1497	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1483  ∃wex 1515  [wsb 1785

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by:  sbco2v  1976  cbvabw  2328