Theorem dfrex2 2627

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)

Assertion

Ref	Expression
dfrex2	⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ)

Proof of Theorem dfrex2

Step	Hyp	Ref	Expression
1		ralnex 2624	. 2 ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ)
2	1	con2bii 322	1 ⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wral 2614  ∃wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2619  df-rex 2620

This theorem is referenced by:  nfrexd  2666  nfrex  2669  rexim  2718  r19.30  2756  r19.35  2758  cbvrexf  2830  rspcimedv  2957  sbcrext  3119  cbvrexcsf  3199  r19.9rzv  3644  rexiunxp  4824  rexxpf  4828  disjex  5823  nnc3n3p1  6278