Theorem ralrimi 2601

Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)

Hypotheses

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

Assertion

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

Proof of Theorem ralrimi

Step	Hyp	Ref	Expression
1		ralrimi.1	. . 3 |- F/ x ph
2		ralrimi.2	. . 3 |- ( ph -> ( x e. A -> ps ) )
3	1, 2	alrimi 1568	. 2 |- ( ph -> A. x ( x e. A -> ps ) )
4		df-ral 2513	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
5	3, 4	sylibr 134	1 |- ( ph -> A. x e. A ps )

Syntax hints:   -> wi 4   A.wal 1393   F/wnf 1506   e. wcel 2200   A.wral 2508

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 1493  ax-gen 1495  ax-4 1556

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513

This theorem is referenced by:  ralrimiv  2602  reximdai  2628  r19.12  2637  rexlimd  2645  rexlimd2  2646  r19.29af2  2671  r19.37  2683  ralidm  3592  iineq2d  3985  mpteq2da  4173  onintonm  4609  mpteqb  5725  fmptdf  5792  eusvobj2  5987  tfri3  6513  mapxpen  7009  fodjuomnilemdc  7311  cc3  7454  zsupcllemstep  10449  fimaxre2  11738  fprodcllemf  12124  fprodap0f  12147  fprodle  12151  bezoutlemmain  12519  bezoutlemzz  12523  exmidunben  12997  mulcncf  15282  limccnp2lem  15350  lfgrnloopen  15931