Theorem alex 1833

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1787. See also the dual pair alnex 1788 / exnal 1834. 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 316	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1826	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		alnex 1788	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	bitri 276	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 207  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 208  df-ex 1787

This theorem is referenced by:  exnal  1834  2nalexn  1835  alimex  1838  emptyal  1915  nfa1  2162  sp  2195  exists2  2666  onvf1odlem1  35338  pm10.253  44813  vk15.4j  44979  vk15.4jVD  45364