Theorem rspe 2599

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 1643	. 2 |- ( ( x e. A /\ ph ) -> E. x ( x e. A /\ ph ) )
2		df-rex 2534	. 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 1545   e. wcel 2209   E.wrex 2529

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 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563

This theorem depends on definitions:  df-bi 117  df-rex 2534

This theorem is referenced by:  rsp2e  2601  ssiun2  4050  tfrlem9  6580  tfrlemibxssdm  6588  tfr1onlembxssdm  6604  tfrcllembxssdm  6617  findcard2  7183  findcard2s  7184  prarloclemup  7852  prmuloc2  7924  ltaddpr  7954  aptiprlemu  7997  cauappcvgprlemopl  8003  cauappcvgprlemopu  8005  cauappcvgprlem2  8017  caucvgprlemopl  8026  caucvgprlemopu  8028  caucvgprlem2  8037  caucvgprprlem2  8067  suplocexprlemrl  8074  suplocexprlemru  8076  suplocexprlemlub  8081