Theorem hbsb 1937

Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)

Hypothesis

Ref	Expression
hbsb.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

Ref	Expression
hbsb	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑦,𝑧

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

Proof of Theorem hbsb

Step	Hyp	Ref	Expression
1		hbsb.1	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nfi 1450	. . 3 ⊢ Ⅎ𝑧𝜑
3	2	nfsb 1934	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
4	3	nfri 1507	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1341  [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:  sb10f  1983  hbsb4  2000  sb8euh  2037  hbab  2156  hblem  2274