Theorem alex 717

Description: Theorem 19.6 of [Margaris] p. 89.

Assertion

Ref	Expression
alex	⊢ (∀xφ ↔ ¬ ∃x ¬ φ)

Proof of Theorem alex

Step	Hyp	Ref	Expression
1		pm4.13 142	. . 3 ⊢ (φ ↔ ¬ ¬ φ)
2	1	bial 695	. 2 ⊢ (∀xφ ↔ ∀x ¬ ¬ φ)
3		alnex 716	. 2 ⊢ (∀x ¬ ¬ φ ↔ ¬ ∃x ¬ φ)
4	2, 3	bitr 151	1 ⊢ (∀xφ ↔ ¬ ∃x ¬ φ)

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

This theorem is referenced by:  exnal 721

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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