Theorem rspe 2554

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 1612	. 2 |- ( ( x e. A /\ ph ) -> E. x ( x e. A /\ ph ) )
2		df-rex 2489	. 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 1514   e. wcel 2175   E.wrex 2484

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 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532

This theorem depends on definitions:  df-bi 117  df-rex 2489

This theorem is referenced by:  rsp2e  2556  ssiun2  3969  tfrlem9  6404  tfrlemibxssdm  6412  tfr1onlembxssdm  6428  tfrcllembxssdm  6441  findcard2  6985  findcard2s  6986  prarloclemup  7607  prmuloc2  7679  ltaddpr  7709  aptiprlemu  7752  cauappcvgprlemopl  7758  cauappcvgprlemopu  7760  cauappcvgprlem2  7772  caucvgprlemopl  7781  caucvgprlemopu  7783  caucvgprlem2  7792  caucvgprprlem2  7822  suplocexprlemrl  7829  suplocexprlemru  7831  suplocexprlemlub  7836