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  4007  tfrlem9  6463  tfrlemibxssdm  6471  tfr1onlembxssdm  6487  tfrcllembxssdm  6500  findcard2  7047  findcard2s  7048  prarloclemup  7678  prmuloc2  7750  ltaddpr  7780  aptiprlemu  7823  cauappcvgprlemopl  7829  cauappcvgprlemopu  7831  cauappcvgprlem2  7843  caucvgprlemopl  7852  caucvgprlemopu  7854  caucvgprlem2  7863  caucvgprprlem2  7893  suplocexprlemrl  7900  suplocexprlemru  7902  suplocexprlemlub  7907