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Theorem wl-sbnf1 33466
 Description: Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2467. Note: This theorem shows that sbnf2 2467 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
Assertion
Ref Expression
wl-sbnf1 (∀𝑥𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Proof of Theorem wl-sbnf1
StepHypRef Expression
1 nf5 2154 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 nfa1 2068 . . 3 𝑥𝑥𝑦𝜑
3 wl-sbhbt 33465 . . 3 (∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
42, 3albid 2128 . 2 (∀𝑥𝑦𝜑 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
51, 4syl5bb 272 1 (∀𝑥𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  Ⅎwnf 1748  [wsb 1937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938 This theorem is referenced by: (None)
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