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Theorem wl-sb8t 37533
Description: Substitution of variable in universal quantifier. Closed form of sb8 2520. (Contributed by Wolf Lammen, 27-Jul-2019.)
Assertion
Ref Expression
wl-sb8t (∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8t
StepHypRef Expression
1 nfa1 2149 . 2 𝑥𝑥𝑦𝜑
2 nfnf1 2152 . . 3 𝑦𝑦𝜑
32nfal 2322 . 2 𝑦𝑥𝑦𝜑
4 sp 2181 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
5 wl-nfs1t 37518 . . 3 (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
65sps 2183 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7 sbequ12 2249 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87a1i 11 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
91, 3, 4, 6, 8cbv2 2406 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wnf 1780  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  wl-sb8et  37534  wl-sbhbt  37535
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