Theorem rspe 2581

Description: Restricted specialization. (Contributed by NM, 12-Oct-1999.)

Assertion

Ref	Expression
rspe	|- ( ( x e. A /\ ph ) -> E. x e. A ph )

Proof of Theorem rspe

Step	Hyp	Ref	Expression
1		19.8a 1638	. 2 |- ( ( x e. A /\ ph ) -> E. x ( x e. A /\ ph ) )
2		df-rex 2516	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
3	1, 2	sylibr 134	1 |- ( ( x e. A /\ ph ) -> E. x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1540   e. wcel 2202   E.wrex 2511

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 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558

This theorem depends on definitions:  df-bi 117  df-rex 2516

This theorem is referenced by:  rsp2e  2583  ssiun2  4013  tfrlem9  6484  tfrlemibxssdm  6492  tfr1onlembxssdm  6508  tfrcllembxssdm  6521  findcard2  7077  findcard2s  7078  prarloclemup  7714  prmuloc2  7786  ltaddpr  7816  aptiprlemu  7859  cauappcvgprlemopl  7865  cauappcvgprlemopu  7867  cauappcvgprlem2  7879  caucvgprlemopl  7888  caucvgprlemopu  7890  caucvgprlem2  7899  caucvgprprlem2  7929  suplocexprlemrl  7936  suplocexprlemru  7938  suplocexprlemlub  7943