Theorem alexn 1579

Description: A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)

Assertion

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

Proof of Theorem alexn

Step	Hyp	Ref	Expression
1		exnal 1574	. . 3 ⊢ (∃y ¬ φ ↔ ¬ ∀yφ)
2	1	albii 1566	. 2 ⊢ (∀x∃y ¬ φ ↔ ∀x ¬ ∀yφ)
3		alnex 1543	. 2 ⊢ (∀x ¬ ∀yφ ↔ ¬ ∃x∀yφ)
4	2, 3	bitri 240	1 ⊢ (∀x∃y ¬ φ ↔ ¬ ∃x∀yφ)

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:  2exnexn  1580