Theorem wl-sbhbt 37598

Description: Closed form of sbhb 2521. Characterizing the expression 𝜑 → ∀𝑥𝜑 using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)

Assertion

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

Proof of Theorem wl-sbhbt

Step	Hyp	Ref	Expression
1		wl-sb8t 37596	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
2	1	imbi2d 340	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
3		19.21t 2209	. . 3 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
4	3	sps 2188	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
5	2, 4	bitr4d 282	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539  Ⅎwnf 1784  [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  ax-6 1968  ax-7 2009  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-sb 2068

This theorem is referenced by:  wl-sbnf1  37599