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Theorem wl-sb8et 32966
Description: Substitution of variable in universal quantifier. Closed form of sb8e 2424. (Contributed by Wolf Lammen, 27-Jul-2019.)
Assertion
Ref Expression
wl-sb8et (∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8et
StepHypRef Expression
1 wl-nfnbi 32946 . . . . 5 (Ⅎ𝑦𝜑 ↔ Ⅎ𝑦 ¬ 𝜑)
21albii 1744 . . . 4 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
3 wl-sb8t 32965 . . . 4 (∀𝑥𝑦 ¬ 𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
42, 3sylbi 207 . . 3 (∀𝑥𝑦𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
5 alnex 1703 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6 sbn 2390 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
76albii 1744 . . . 4 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
8 alnex 1703 . . . 4 (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
97, 8bitri 264 . . 3 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
104, 5, 93bitr3g 302 . 2 (∀𝑥𝑦𝜑 → (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑))
1110con4bid 307 1 (∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478  wex 1701  wnf 1705  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  wl-mo3t  32990  wl-sb8mot  32992
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