Theorem alex 1829

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1784. See also the dual pair alnex 1785 / 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

Step	Hyp	Ref	Expression
1		notnotb 314	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1823	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		alnex 1785	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	bitri 274	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  exnal  1830  2nalexn  1831  alimex  1834  emptyal  1912  nfa1  2150  sp  2178  exists2  2663  pm10.253  41869  vk15.4j  42037  vk15.4jVD  42423