Theorem rexlimd 2546

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 2503	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexlimd.2	. . 3 |- F/ x ch
5	4	r19.23 2540	. 2 |- ( A. x e. A ( ps -> ch ) <-> ( E. x e. A ps -> ch ) )
6	3, 5	sylib 121	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   F/wnf 1436   e. wcel 1480   A.wral 2416   E.wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421  df-rex 2422

This theorem is referenced by:  rexlimdv  2548  ralxfrALT  4388  fvmptt  5512  ffnfv  5578  nneneq  6751  ac6sfi  6792  prarloclem3step  7304  prmuloc2  7375  caucvgprprlemaddq  7516  axpre-suploclemres  7709  lbzbi  9408  divalglemeunn  11618  divalglemeuneg  11620  oddpwdclemdvds  11848  oddpwdclemndvds  11849