Theorem rspe 2591

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 1639	. 2 |- ( ( x e. A /\ ph ) -> E. x ( x e. A /\ ph ) )
2		df-rex 2526	. 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 1541   e. wcel 2203   E.wrex 2521

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 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559

This theorem depends on definitions:  df-bi 117  df-rex 2526

This theorem is referenced by:  rsp2e  2593  ssiun2  4034  tfrlem9  6550  tfrlemibxssdm  6558  tfr1onlembxssdm  6574  tfrcllembxssdm  6587  findcard2  7146  findcard2s  7147  prarloclemup  7810  prmuloc2  7882  ltaddpr  7912  aptiprlemu  7955  cauappcvgprlemopl  7961  cauappcvgprlemopu  7963  cauappcvgprlem2  7975  caucvgprlemopl  7984  caucvgprlemopu  7986  caucvgprlem2  7995  caucvgprprlem2  8025  suplocexprlemrl  8032  suplocexprlemru  8034  suplocexprlemlub  8039