Theorem hbs1f 1664

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 1659	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
3	2, 1	hbxfrbi 1361	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1241  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by: (None)