Theorem wl-sb8ft 38233

Description: Substitution of variable in universal quantifier. Closed form of sb8f 2385. (Contributed by Wolf Lammen, 27-Apr-2025.)

Assertion

Ref	Expression
wl-sb8ft	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem wl-sb8ft

Step	Hyp	Ref	Expression
1		sbft 2304	. . . 4 ⊢ (Ⅎ𝑦𝜑 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
2	1	alimi 1840	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3		albi 1847	. . 3 ⊢ (∀𝑥([𝑥 / 𝑦]𝜑 ↔ 𝜑) → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥𝜑))
4	2, 3	syl 18	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥𝜑))
5		wl-sb9v 38232	. 2 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
6	4, 5	bitr3di 289	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  Ⅎwnf 1812  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-11 2191  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813  df-sb 2096

This theorem is used by:  wl-sb8eft  38234