Theorem rexlimd 2619

Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
rexlimd.1	|- F/ x ph
rexlimd.2	|- F/ x ch
rexlimd.3	|- ( ph -> ( x e. A -> ( ps -> ch ) ) )

Assertion

Ref	Expression
rexlimd	|- ( ph -> ( E. x e. A ps -> ch ) )

Proof of Theorem rexlimd

Step	Hyp	Ref	Expression
1		rexlimd.1	. . 3 |- F/ x ph
2		rexlimd.3	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	1, 2	ralrimi 2576	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexlimd.2	. . 3 |- F/ x ch
5	4	r19.23 2613	. 2 |- ( A. x e. A ( ps -> ch ) <-> ( E. x e. A ps -> ch ) )
6	3, 5	sylib 122	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   F/wnf 1482   e. wcel 2175   A.wral 2483   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-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-ral 2488  df-rex 2489

This theorem is referenced by:  rexlimdv  2621  ralxfrALT  4513  fvmptt  5670  ffnfv  5737  nneneq  6953  ac6sfi  6994  prarloclem3step  7608  prmuloc2  7679  caucvgprprlemaddq  7820  axpre-suploclemres  8013  lbzbi  9736  divalglemeunn  12203  divalglemeuneg  12205  oddpwdclemdvds  12463  oddpwdclemndvds  12464  trirec0  15945