Theorem rexlimd 2474

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 2432	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexlimd.2	. . 3 |- F/ x ch
5	4	r19.23 2468	. 2 |- ( A. x e. A ( ps -> ch ) <-> ( E. x e. A ps -> ch ) )
6	3, 5	sylib 120	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   F/wnf 1389   e. wcel 1433   A.wral 2348   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  rexlimdv  2476  ralxfrALT  4217  fvmptt  5283  ffnfv  5344  nneneq  6343  ac6sfi  6379  prarloclem3step  6686  prmuloc2  6757  caucvgprprlemaddq  6898  lbzbi  8701  divalglemeunn  10321  divalglemeuneg  10323  oddpwdclemdvds  10548  oddpwdclemndvds  10549