Theorem ralrimd 2959

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)

Hypotheses

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

Assertion

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

Proof of Theorem ralrimd

Step	Hyp	Ref	Expression
1		ralrimd.1	. . 3 |- F/ x ph
2		ralrimd.2	. . 3 |- F/ x ps
3		ralrimd.3	. . 3 |- ( ph -> ( ps -> ( x e. A -> ch ) ) )
4	1, 2, 3	alrimd 2084	. 2 |- ( ph -> ( ps -> A. x ( x e. A -> ch ) ) )
5		df-ral 2917	. 2 |- ( A. x e. A ch <-> A. x ( x e. A -> ch ) )
6	4, 5	syl6ibr 242	1 |- ( ph -> ( ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710  df-ral 2917

This theorem is referenced by:  reusv2lem3  4871  fliftfun  6562  mapxpen  8126  domtriomlem  9264  dedekind  10200  fzrevral  12425  matunitlindflem2  33406  riotasv3d  34246  ssralv2  38737  setrec1lem2  42435