Theorem alex 1753

Description: Universal quantifier in terms of existential quantifier and negation. 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 304	. . 3 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
2	1	albii 1747	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥 ¬ ¬ 𝜑)
3		alnex 1706	. 2 ⊢ (∀𝑥 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
4	2, 3	bitri 264	1 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  exnal  1754  2nalexn  1755  alimex  1758  19.3v  1897  nfa1  2028  sp  2053  exists2  2562  19.9alt  34252  pm10.253  38561  vk15.4j  38734  vk15.4jVD  39150