Theorem rsp2e 2557

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 1000	. . 3 |- ( ( x e. A /\ y e. B /\ ph ) -> x e. A )
2		rspe 2555	. . . 4 |- ( ( y e. B /\ ph ) -> E. y e. B ph )
3	2	3adant1 1018	. . 3 |- ( ( x e. A /\ y e. B /\ ph ) -> E. y e. B ph )
4		19.8a 1613	. . 3 |- ( ( x e. A /\ E. y e. B ph ) -> E. x ( x e. A /\ E. y e. B ph ) )
5	1, 3, 4	syl2anc 411	. 2 |- ( ( x e. A /\ y e. B /\ ph ) -> E. x ( x e. A /\ E. y e. B ph ) )
6		df-rex 2490	. 2 |- ( E. x e. A E. y e. B ph <-> E. x ( x e. A /\ E. y e. B ph ) )
7	5, 6	sylibr 134	1 |- ( ( x e. A /\ y e. B /\ ph ) -> E. x e. A E. y e. B ph )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   E.wex 1515   e. wcel 2176   E.wrex 2485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533

This theorem depends on definitions:  df-bi 117  df-3an 983  df-rex 2490

This theorem is referenced by: (None)