Theorem wl-sb8ft 37497

Description: Substitution of variable in universal quantifier. Closed form of sb8f 2359. (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 2271	. . . 4 ⊢ (Ⅎ𝑦𝜑 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
2	1	alimi 1809	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3		albi 1816	. . 3 ⊢ (∀𝑥([𝑥 / 𝑦]𝜑 ↔ 𝜑) → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥𝜑))
4	2, 3	syl 17	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑥𝜑))
5		wl-sb9v 37496	. 2 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
6	4, 5	bitr3di 286	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  Ⅎwnf 1781  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by:  wl-sb8eft  37498