Theorem hbsbw 2172

Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Version of hbsb 2532 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993.) (Revised by GG, 23-May-2024.) (Proof shortened by Wolf Lammen, 14-May-2025.)

Hypothesis

Ref	Expression
hbsbw.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem hbsbw

Step	Hyp	Ref	Expression
1		hbsbw.1	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	sbimi 2074	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑)
3		sbal 2170	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
4	2, 3	sylib 218	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-11 2158

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-sb 2065

This theorem is referenced by:  nfsbv  2334  hbab  2728  hblem  2876