Theorem nfsb 1997

Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
nfsb.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfsb	⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Distinct variable group:   𝑦,𝑧

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

Proof of Theorem nfsb

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsb.1	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	nfsbxy 1993	. . 3 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑
3	2	nfsbxy 1993	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4		ax-17 1572	. . . 4 ⊢ (𝜑 → ∀𝑤𝜑)
5	4	sbco2vh 1996	. . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6	5	nfbii 1519	. 2 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7	3, 6	mpbi 145	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1506  [wsb 1808

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  hbsb  2000  sbco2yz  2014  sbcomxyyz  2023  hbsbd  2033  nfsb4or  2072  sb8eu  2090  nfeu  2096  cbvab  2353  cbvralf  2756  cbvrexf  2757  cbvreu  2763  cbvralsv  2781  cbvrexsv  2782  cbvrab  2797  cbvreucsf  3189  cbvrabcsf  3190  cbvopab1  4156  cbvmptf  4177  cbvmpt  4178  ralxpf  4867  rexxpf  4868  cbviota  5282  sb8iota  5285  cbvriota  5965  dfoprab4f  6337