Theorem dfral2 2627

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

Assertion

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

Proof of Theorem dfral2

Step	Hyp	Ref	Expression
1		rexnal 2626	. 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 2615  ∃wrex 2616

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 2620  df-rex 2621

This theorem is referenced by:  dfpss4  3889  ncfinraiselem2  4481  evenodddisjlem1  4516  nnadjoinlem1  4520  nnpweqlem1  4523  tfinnnlem1  4534  spfinex  4538  extex  5916