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Theorem alex 1825
Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1779. See also the dual pair alnex 1780 / exnal 1826. Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
Assertion
Ref Expression
alex (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Proof of Theorem alex
StepHypRef Expression
1 notnotb 315 . . 3 (𝜑 ↔ ¬ ¬ 𝜑)
21albii 1818 . 2 (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3 alnex 1780 . 2 (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
42, 3bitri 275 1 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wal 1537  wex 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808
This theorem depends on definitions:  df-bi 207  df-ex 1779
This theorem is referenced by:  exnal  1826  2nalexn  1827  alimex  1830  emptyal  1907  nfa1  2150  sp  2182  exists2  2661  pm10.253  44386  vk15.4j  44553  vk15.4jVD  44939
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