Theorem wl-sbhbt 34340

Description: Closed form of sbhb 2517. 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 34338	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
2	1	imbi2d 342	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
3		19.21t 2171	. . 3 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
4	3	sps 2148	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
5	2, 4	bitr4d 283	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1520  Ⅎwnf 1765  [wsb 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043

This theorem is referenced by:  wl-sbnf1  34341