Theorem rsp2e 2678

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 955	. . 3 |- ((x e. A /\ y e. B /\ ph) -> x e. A)
2		rspe 2676	. . . 4 |- ((y e. B /\ ph) -> E.y e. B ph)
3	2	3adant1 973	. . 3 |- ((x e. A /\ y e. B /\ ph) -> E.y e. B ph)
4		19.8a 1756	. . 3 |- ((x e. A /\ E.y e. B ph) -> E.x(x e. A /\ E.y e. B ph))
5	1, 3, 4	syl2anc 642	. 2 |- ((x e. A /\ y e. B /\ ph) -> E.x(x e. A /\ E.y e. B ph))
6		df-rex 2621	. 2 |- (E.x e. A E.y e. B ph <-> E.x(x e. A /\ E.y e. B ph))
7	5, 6	sylibr 203	1 |- ((x e. A /\ y e. B /\ ph) -> E.x e. A E.y e. B ph)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  E.wex 1541   e. wcel 1710  E.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  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-rex 2621

This theorem is referenced by: (None)