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Theorem alex 1817
Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1772. See also the dual pair alnex 1773 / exnal 1818. 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 316 . . 3 (𝜑 ↔ ¬ ¬ 𝜑)
21albii 1811 . 2 (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3 alnex 1773 . 2 (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
42, 3bitri 276 1 (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wal 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-ex 1772
This theorem is referenced by:  exnal  1818  2nalexn  1819  alimex  1822  emptyal  1900  19.3vOLD  1980  nfa1  2146  sp  2172  exists2  2742  pm10.253  40571  vk15.4j  40739  vk15.4jVD  41125
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