Theorem hbsbw 2184

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 2534 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993.) Remove dependencies on axioms. (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 2086	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑧𝜑)
3		sbal 2182	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
4	2, 3	sylib 220	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1546  [wsb 2074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-11 2170

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075

This theorem is referenced by:  nfsbv  2341  hbab  2729  hblem  2872