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Theorem wl-sb8et 37534
Description: Substitution of variable in universal quantifier. Closed form of sb8e 2521. (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 nfnbi 1852 . . . . 5 (Ⅎ𝑦𝜑 ↔ Ⅎ𝑦 ¬ 𝜑)
21albii 1816 . . . 4 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
3 wl-sb8t 37533 . . . 4 (∀𝑥𝑦 ¬ 𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
42, 3sylbi 217 . . 3 (∀𝑥𝑦𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
5 alnex 1778 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6 sbn 2279 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
76albii 1816 . . . 4 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
8 alnex 1778 . . . 4 (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
97, 8bitri 275 . . 3 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
104, 5, 93bitr3g 313 . 2 (∀𝑥𝑦𝜑 → (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑))
1110con4bid 317 1 (∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wal 1535  wex 1776  wnf 1780  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063
This theorem is referenced by:  wl-sb8mot  37561
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