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Theorem alex 1827
 Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1782. See also the dual pair alnex 1783 / exnal 1828. 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 318 . . 3 (𝜑 ↔ ¬ ¬ 𝜑)
21albii 1821 . 2 (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3 alnex 1783 . 2 (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
42, 3bitri 278 1 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  exnal  1828  2nalexn  1829  alimex  1832  emptyal  1909  19.3vOLD  1989  nfa1  2152  sp  2180  exists2  2724  pm10.253  41109  vk15.4j  41277  vk15.4jVD  41663
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