Theorem ralrimd 2548

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 1603	. 2 |- ( ph -> ( ps -> A. x ( x e. A -> ch ) ) )
5		df-ral 2453	. 2 |- ( A. x e. A ch <-> A. x ( x e. A -> ch ) )
6	4, 5	syl6ibr 161	1 |- ( ph -> ( ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   A.wal 1346   F/wnf 1453   e. wcel 2141   A.wral 2448

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 1440  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453

This theorem is referenced by:  ralrimdv  2549  fliftfun  5775  mapxpen  6826  fzrevral  10061