Theorem 2nalexn 1573

Description: Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

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

Proof of Theorem 2nalexn

Step	Hyp	Ref	Expression
1		df-ex 1542	. . 3 ⊢ (∃x∃y ¬ φ ↔ ¬ ∀x ¬ ∃y ¬ φ)
2		alex 1572	. . . 4 ⊢ (∀yφ ↔ ¬ ∃y ¬ φ)
3	2	albii 1566	. . 3 ⊢ (∀x∀yφ ↔ ∀x ¬ ∃y ¬ φ)
4	1, 3	xchbinxr 302	. 2 ⊢ (∃x∃y ¬ φ ↔ ¬ ∀x∀yφ)
5	4	bicomi 193	1 ⊢ (¬ ∀x∀yφ ↔ ∃x∃y ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  spc2gv  2942