Theorem rexlimd 2544

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 2501	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexlimd.2	. . 3 |- F/ x ch
5	4	r19.23 2538	. 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 2414   E.wrex 2415

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 2419  df-rex 2420

This theorem is referenced by:  rexlimdv  2546  ralxfrALT  4383  fvmptt  5505  ffnfv  5571  nneneq  6744  ac6sfi  6785  prarloclem3step  7297  prmuloc2  7368  caucvgprprlemaddq  7509  axpre-suploclemres  7702  lbzbi  9401  divalglemeunn  11607  divalglemeuneg  11609  oddpwdclemdvds  11837  oddpwdclemndvds  11838