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Theorem wl-sb8et 34230
Description: Substitution of variable in universal quantifier. Closed form of sb8e 2484. (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 1817 . . . . 5 (Ⅎ𝑦𝜑 ↔ Ⅎ𝑦 ¬ 𝜑)
21albii 1782 . . . 4 (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
3 wl-sb8t 34229 . . . 4 (∀𝑥𝑦 ¬ 𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
42, 3sylbi 209 . . 3 (∀𝑥𝑦𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
5 alnex 1744 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6 sbn 2214 . . . . 5 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
76albii 1782 . . . 4 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
8 alnex 1744 . . . 4 (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
97, 8bitri 267 . . 3 (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
104, 5, 93bitr3g 305 . 2 (∀𝑥𝑦𝜑 → (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑))
1110con4bid 309 1 (∀𝑥𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wal 1505  wex 1742  wnf 1746  [wsb 2015
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 1965  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-nf 1747  df-sb 2016
This theorem is referenced by:  wl-mo3t  34253  wl-sb8mot  34255
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