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Theorem wl-sbnf1 36117
Description: Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2354. Note: This theorem shows that sbnf2 2354 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 2278 . 2 (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2 nfa1 2148 . . 3 𝑥𝑥𝑦𝜑
3 wl-sbhbt 36116 . . 3 (∀𝑥𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
42, 3albid 2215 . 2 (∀𝑥𝑦𝜑 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
51, 4bitrid 282 1 (∀𝑥𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wnf 1785  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068
This theorem is referenced by: (None)
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