Theorem wl-cbvalsbi 37590

Description: Change bounded variables in a special case. The reverse direction seems to involve ax-11 2160. My hope is that I will in some future be able to prove mo3 2559 with reversed quantifiers not using ax-11 2160. See also the remark in mo4 2561, which lead me to this effort. (Contributed by Wolf Lammen, 5-Mar-2024.)

Assertion

Ref	Expression
wl-cbvalsbi	⊢ (∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-cbvalsbi

Step	Hyp	Ref	Expression
1		stdpc4 2071	. 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2	1	alrimiv 1928	1 ⊢ (∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1539  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-sb 2068

This theorem is referenced by: (None)