Theorem alexn 1848

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 1830	. . 3 ⊢ (∃𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦𝜑)
2	1	albii 1823	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ∀𝑥 ¬ ∀𝑦𝜑)
3		alnex 1785	. 2 ⊢ (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)
4	2, 3	bitri 274	1 ⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  2exnexn  1849  nalset  5232  kmlem2  9838