Theorem nfsb 1934

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 1930	. . 3 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑
3	2	nfsbxy 1930	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4		ax-17 1514	. . . 4 ⊢ (𝜑 → ∀𝑤𝜑)
5	4	sbco2vh 1933	. . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6	5	nfbii 1461	. 2 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7	3, 6	mpbi 144	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1448  [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  hbsb  1937  sbco2yz  1951  sbcomxyyz  1960  hbsbd  1970  nfsb4or  2009  sb8eu  2027  nfeu  2033  cbvab  2290  cbvralf  2685  cbvrexf  2686  cbvreu  2690  cbvralsv  2708  cbvrexsv  2709  cbvrab  2724  cbvreucsf  3109  cbvrabcsf  3110  cbvopab1  4055  cbvmptf  4076  cbvmpt  4077  ralxpf  4750  rexxpf  4751  cbviota  5158  sb8iota  5160  cbvriota  5808  dfoprab4f  6161