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Theorem wl-dfrexv 34972
Description: Alternate definition of restricted universal quantification (df-wl-rex 34969) when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 25-May-2023.)
Assertion
Ref Expression
wl-dfrexv (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfrexv
StepHypRef Expression
1 wl-dfralv 34964 . . 3 (∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
21notbii 323 . 2 (¬ ∀(𝑥 : 𝐴) ¬ 𝜑 ↔ ¬ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
3 df-wl-rex 34969 . 2 (∃(𝑥 : 𝐴)𝜑 ↔ ¬ ∀(𝑥 : 𝐴) ¬ 𝜑)
4 exnalimn 1845 . 2 (∃𝑥(𝑥𝐴𝜑) ↔ ¬ ∀𝑥(𝑥𝐴 → ¬ 𝜑))
52, 3, 43bitr4i 306 1 (∃(𝑥 : 𝐴)𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536  wex 1781  wcel 2114  wl-ral 34954  wl-rex 34955
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-8 2116  ax-11 2161
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clel 2894  df-wl-ral 34959  df-wl-rex 34969
This theorem is referenced by:  wl-dfreuv  34981
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