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Theorem alex 1901
Description: Universal quantifier in terms of existential quantifier and negation. 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 304 . . 3 (𝜑 ↔ ¬ ¬ 𝜑)
21albii 1895 . 2 (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3 alnex 1854 . 2 (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
42, 3bitri 264 1 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  exnal  1902  2nalexn  1903  alimex  1906  19.3v  2066  nfa1  2184  sp  2207  exists2  2711  19.9alt  34774  pm10.253  39087  vk15.4j  39259  vk15.4jVD  39672
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