Theorem 2exnaln 1862

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 1813	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥 ¬ ∃𝑦𝜑)
2		alnex 1814	. . 3 ⊢ (∀𝑦 ¬ 𝜑 ↔ ¬ ∃𝑦𝜑)
3	2	albii 1852	. 2 ⊢ (∀𝑥∀𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑦𝜑)
4	1, 3	xchbinxr 338	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  2nexaln  1863  excomimw  2077  exexw  2086  excom  2200  opab0  5544  bj-modal4e  37383