Theorem alexn 726

Description: A relationship between two quantifiers and negation.

Assertion

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

Proof of Theorem alexn

Step	Hyp	Ref	Expression
1		exnal 721	. . 3 ⊢ (∃y ¬ φ ↔ ¬ ∀yφ)
2	1„/TD>	bial 695	. 2 ⊢ (∀x∃y ¬ φ ↔ ∀x ¬ ∀yφ)
3		alnex 716	. 2 ⊢ (∀x ¬ ∀yφ ↔ ¬ ∃x∀yφ)
4	2, 3	bitr 151	1 ⊢ (∀x∃y ¬ φ ↔ ¬ ∃x∀yφ)

Syntax hints:  ¬ wn 1   ↔ wb 127  ∀wal 672  ∃wex 678

This theorem is referenced by:  nalset 1482  kmlem2 3581

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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