Theorem rspe 2579

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 1636	. 2 |- ( ( x e. A /\ ph ) -> E. x ( x e. A /\ ph ) )
2		df-rex 2514	. 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 1538   e. wcel 2200   E.wrex 2509

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 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556

This theorem depends on definitions:  df-bi 117  df-rex 2514

This theorem is referenced by:  rsp2e  2581  ssiun2  4011  tfrlem9  6480  tfrlemibxssdm  6488  tfr1onlembxssdm  6504  tfrcllembxssdm  6517  findcard2  7071  findcard2s  7072  prarloclemup  7705  prmuloc2  7777  ltaddpr  7807  aptiprlemu  7850  cauappcvgprlemopl  7856  cauappcvgprlemopu  7858  cauappcvgprlem2  7870  caucvgprlemopl  7879  caucvgprlemopu  7881  caucvgprlem2  7890  caucvgprprlem2  7920  suplocexprlemrl  7927  suplocexprlemru  7929  suplocexprlemlub  7934