Theorem wl-cbvalsbi 38229

Description: Change bounded variables in a special case. The reverse direction seems to involve ax-11 2192. My hope is that I will in some future be able to prove mo3 2592 with reversed quantifiers not using ax-11 2192. See also the remark in mo4 2594, 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 2102	. 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2	1	alrimiv 1957	1 ⊢ (∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  [wsb 2096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097

This theorem is used by: (None)