Theorem rspe 2546

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 1604	. 2 |- ( ( x e. A /\ ph ) -> E. x ( x e. A /\ ph ) )
2		df-rex 2481	. 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 1506   e. wcel 2167   E.wrex 2476

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 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524

This theorem depends on definitions:  df-bi 117  df-rex 2481

This theorem is referenced by:  rsp2e  2548  ssiun2  3959  tfrlem9  6377  tfrlemibxssdm  6385  tfr1onlembxssdm  6401  tfrcllembxssdm  6414  findcard2  6950  findcard2s  6951  prarloclemup  7562  prmuloc2  7634  ltaddpr  7664  aptiprlemu  7707  cauappcvgprlemopl  7713  cauappcvgprlemopu  7715  cauappcvgprlem2  7727  caucvgprlemopl  7736  caucvgprlemopu  7738  caucvgprlem2  7747  caucvgprprlem2  7777  suplocexprlemrl  7784  suplocexprlemru  7786  suplocexprlemlub  7791