Theorem rexlimd 2648

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 2604	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexlimd.2	. . 3 |- F/ x ch
5	4	r19.23 2642	. 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 1509   e. wcel 2202   A.wral 2511   E.wrex 2512

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  rexlimdv  2650  ralxfrALT  4570  fvmptt  5747  ffnfv  5813  nneneq  7086  ac6sfi  7130  prarloclem3step  7759  prmuloc2  7830  caucvgprprlemaddq  7971  axpre-suploclemres  8164  lbzbi  9894  reuccatpfxs1  11377  divalglemeunn  12545  divalglemeuneg  12547  oddpwdclemdvds  12805  oddpwdclemndvds  12806  trirec0  16759