Theorem alex 1823

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1777. See also the dual pair alnex 1778 / exnal 1824. 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 315	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1816	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		alnex 1778	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	bitri 275	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1535  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 207  df-ex 1777

This theorem is referenced by:  exnal  1824  2nalexn  1825  alimex  1828  emptyal  1906  nfa1  2149  sp  2181  exists2  2660  pm10.253  44358  vk15.4j  44526  vk15.4jVD  44912