Theorem rsp2e 2517

Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.)

Assertion

Ref	Expression
rsp2e	|- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph )

Proof of Theorem rsp2e

Step	Hyp	Ref	Expression
1		simp1 987	. . 3 |- ( ( x e. A /\ y e. B /\ ph ) -> x e. A )
2		rspe 2515	. . . 4 |- ( ( y e. B /\ ph ) -> E. y e. B ph )
3	2	3adant1 1005	. . 3 |- ( ( x e. A /\ y e. B /\ ph ) -> E. y e. B ph )
4		19.8a 1578	. . 3 |- ( ( x e. A /\ E. y e. B ph ) -> E. x ( x e. A /\ E. y e. B ph ) )
5	1, 3, 4	syl2anc 409	. 2 |- ( ( x e. A /\ y e. B /\ ph ) -> E. x ( x e. A /\ E. y e. B ph ) )
6		df-rex 2450	. 2 |- ( E. x e. A E. y e. B ph <-> E. x ( x e. A /\ E. y e. B ph ) )
7	5, 6	sylibr 133	1 |- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   E.wex 1480   e. wcel 2136   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-3an 970  df-rex 2450

This theorem is referenced by: (None)