Theorem hbs1f 1704

Description: If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
hbs1f.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbs1f	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem hbs1f

Step	Hyp	Ref	Expression
1		hbs1f.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	sbh 1699	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
3	2, 1	hbxfrbi 1401	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1282  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by: (None)