Theorem nfsb 1946

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 1942	. . 3 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑
3	2	nfsbxy 1942	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4		ax-17 1526	. . . 4 ⊢ (𝜑 → ∀𝑤𝜑)
5	4	sbco2vh 1945	. . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6	5	nfbii 1473	. 2 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7	3, 6	mpbi 145	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1460  [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  hbsb  1949  sbco2yz  1963  sbcomxyyz  1972  hbsbd  1982  nfsb4or  2021  sb8eu  2039  nfeu  2045  cbvab  2301  cbvralf  2697  cbvrexf  2698  cbvreu  2702  cbvralsv  2720  cbvrexsv  2721  cbvrab  2736  cbvreucsf  3122  cbvrabcsf  3123  cbvopab1  4077  cbvmptf  4098  cbvmpt  4099  ralxpf  4774  rexxpf  4775  cbviota  5184  sb8iota  5186  cbvriota  5841  dfoprab4f  6194