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Theorem wl-sb8t 34946
Description: Substitution of variable in universal quantifier. Closed form of sb8 2539. (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 2153 . 2 𝑥𝑥𝑦𝜑
2 nfnf1 2156 . . 3 𝑦𝑦𝜑
32nfal 2334 . 2 𝑦𝑥𝑦𝜑
4 sp 2181 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
5 wl-nfs1t 34935 . . 3 (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
65sps 2183 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7 sbequ12 2252 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87a1i 11 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
91, 3, 4, 6, 8cbv2 2415 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536  wnf 1785  [wsb 2069
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 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070
This theorem is referenced by:  wl-sb8et  34947  wl-sbhbt  34948
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