Theorem dfral2 1211

Description: Relationship between restricted universal and existential quantifiers.

Assertion

Ref	Expression
dfral2	⊢ (∀x ∈ A φ ↔ ¬ ∃x ∈ A ¬ φ)

Proof of Theorem dfral2

Step	Hyp	Ref	Expression
1		rexnal 1210	. 2 ⊢ (∃x ∈ A ¬ φ ↔ ¬ ∀x ∈ A φ)
2	1	bicon2i 194	1 ⊢ (∀x ∈ A φ ↔ ¬ ∃x ∈ A ¬ φ)

Syntax hints:  ¬ wn 1   ↔ wb 127  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  indstr 4611  str 5698

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  df-ral 1205  df-rex 1206

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