Metamath Proof Explorer


Theorem r2ex

Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020)

Ref Expression
Assertion r2ex
|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )

Proof

Step Hyp Ref Expression
1 r2al
 |-  ( A. x e. A A. y e. B -. ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> -. ph ) )
2 1 r2exlem
 |-  ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )