Theorem alex 1788

Description: Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1743. See also the dual pair alnex 1744 / exnal 1789. 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 307	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1782	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		alnex 1744	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	bitri 267	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772

This theorem depends on definitions:  df-bi 199  df-ex 1743

This theorem is referenced by:  exnal  1789  2nalexn  1790  alimex  1793  19.3v  1939  nfa1  2088  sp  2111  exists2  2695  exists2OLD  2696  wl-nax6al  34200  pm10.253  40110  vk15.4j  40281  vk15.4jVD  40667