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Theorem alex 1829
Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1783. See also the dual pair alnex 1784 / exnal 1830. 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 1822 . 2 (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3 alnex 1784 . 2 (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
42, 3bitri 275 1 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1540  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-ex 1783
This theorem is referenced by:  exnal  1830  2nalexn  1831  alimex  1834  emptyal  1912  nfa1  2149  sp  2177  exists2  2662  pm10.253  42716  vk15.4j  42884  vk15.4jVD  43270
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