Theorem alexn 1841

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 1823	. . 3 ⊢ (∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦𝜑)
2	1	albii 1816	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∀𝑦𝜑)
3		alnex 1778	. 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)
4	2, 3	bitri 277	1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by:  2exnexn  1842  nalset  5210  kmlem2  9571