Theorem nfsb 1962

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 1958	. . 3 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑
3	2	nfsbxy 1958	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑
4		ax-17 1537	. . . 4 ⊢ (𝜑 → ∀𝑤𝜑)
5	4	sbco2vh 1961	. . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
6	5	nfbii 1484	. 2 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
7	3, 6	mpbi 145	1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑

Syntax hints:  Ⅎwnf 1471  [wsb 1773

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  hbsb  1965  sbco2yz  1979  sbcomxyyz  1988  hbsbd  1998  nfsb4or  2037  sb8eu  2055  nfeu  2061  cbvab  2317  cbvralf  2718  cbvrexf  2719  cbvreu  2724  cbvralsv  2742  cbvrexsv  2743  cbvrab  2758  cbvreucsf  3145  cbvrabcsf  3146  cbvopab1  4102  cbvmptf  4123  cbvmpt  4124  ralxpf  4808  rexxpf  4809  cbviota  5220  sb8iota  5222  cbvriota  5884  dfoprab4f  6246