Mathbox for Wolf Lammen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-sb8t Structured version   Visualization version   GIF version

Theorem wl-sb8t 33661
 Description: Substitution of variable in universal quantifier. Closed form of sb8 2570. (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 2183 . 2 𝑥𝑥𝑦𝜑
2 nfnf1 2186 . . 3 𝑦𝑦𝜑
32nfal 2316 . 2 𝑦𝑥𝑦𝜑
4 sp 2206 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜑)
5 wl-nfs1t 33652 . . 3 (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
65sps 2208 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7 sbequ12 2266 . . 3 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
87a1i 11 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
91, 3, 4, 6, 8cbv2 2430 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1628  Ⅎwnf 1855  [wsb 2048 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ex 1852  df-nf 1857  df-sb 2049 This theorem is referenced by:  wl-sb8et  33662  wl-sbhbt  33663
 Copyright terms: Public domain W3C validator