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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 (∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Proof of Theorem wl-sbhbt
StepHypRef Expression
1 wl-sb8t 34338 . . 3 (∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
21imbi2d 342 . 2 (∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
3 19.21t 2171 . . 3 (Ⅎ𝑦𝜑 → (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
43sps 2148 . 2 (∀𝑥𝑦𝜑 → (∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)))
52, 4bitr4d 283 1 (∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1520  wnf 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
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