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Theorem wl-sb8t 33313
Description: Substitution of variable in universal quantifier. Closed form of sb8 2423. (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 2027 . 2 𝑥𝑥𝑦𝜑
2 nfnf1 2030 . . 3 𝑦𝑦𝜑
32nfal 2152 . 2 𝑦𝑥𝑦𝜑
4 sp 2052 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
5 wl-nfs1t 33304 . . 3 (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
65sps 2054 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7 sbequ12 2110 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87a1i 11 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
91, 3, 4, 6, 8cbv2 2269 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1480  wnf 1707  [wsb 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1704  df-nf 1709  df-sb 1880
This theorem is referenced by:  wl-sb8et  33314  wl-sbhbt  33315
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