Theorem 2exnaln 1830

Description: Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
2exnaln	⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)

Proof of Theorem 2exnaln

Step	Hyp	Ref	Expression
1		df-ex 1781	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦𝜑)
2		alnex 1782	. . 3 ⊢ (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑)
3	2	albii 1820	. 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑)
4	1, 3	xchbinxr 334	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1538  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 206  df-ex 1781

This theorem is referenced by:  2nexaln  1831  exexw  2053  excom  2161  cgsex4g  3485  opab0  5485  bj-modal4e  34958