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Theorem wl-sb8t 34165
Description: Substitution of variable in universal quantifier. Closed form of sb8 2481. (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 2086 . 2 𝑥𝑥𝑦𝜑
2 nfnf1 2089 . . 3 𝑦𝑦𝜑
32nfal 2261 . 2 𝑦𝑥𝑦𝜑
4 sp 2109 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
5 wl-nfs1t 34155 . . 3 (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
65sps 2111 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7 sbequ12 2177 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87a1i 11 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
91, 3, 4, 6, 8cbv2 2334 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1505  wnf 1746  [wsb 2013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747  df-sb 2014
This theorem is referenced by:  wl-sb8et  34166  wl-sbhbt  34167
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