Theorem alex 1845

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1799. See also the dual pair alnex 1800 / exnal 1846. 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 317	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1838	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		alnex 1800	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	bitri 277	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828

This theorem depends on definitions:  df-bi 209  df-ex 1799

This theorem is referenced by:  exnal  1846  2nalexn  1847  alimex  1850  emptyal  1927  nfa1  2184  sp  2217  exists2  2687  onvf1odlem1  35410  pm10.253  44902  vk15.4j  45068  vk15.4jVD  45453