Theorem hbsbw 2350

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 2566 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)

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	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	nf5i 2149	. . 3 ⊢ Ⅎ𝑧𝜑
3	2	nfsbv 2348	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
4	3	nf5ri 2194	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1534  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by:  hbab  2809  hblem  2942