Theorem alex 1858

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1812. See also the dual pair alnex 1813 / exnal 1859. 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 318	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1851	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		alnex 1813	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	bitri 278	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  exnal  1859  2nalexn  1860  alimex  1863  emptyal  1940  nfa1  2187  sp  2219  exists2  2686  onvf1odlem1  35951  pm10.253  45440  vk15.4j  45605  vk15.4jVD  45990