Theorem hbs1f 1774

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 1769	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
3	2, 1	hbxfrbi 1465	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1346  [wsb 1755

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-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by: (None)