Theorem ralrimi 2565

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 1533	. 2 |- ( ph -> A. x ( x e. A -> ps ) )
4		df-ral 2477	. 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 1362   F/wnf 1471   e. wcel 2164   A.wral 2472

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 1458  ax-gen 1460  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2477

This theorem is referenced by:  ralrimiv  2566  reximdai  2592  r19.12  2600  rexlimd  2608  rexlimd2  2609  r19.29af2  2634  r19.37  2646  ralidm  3548  iineq2d  3933  mpteq2da  4119  onintonm  4550  mpteqb  5649  fmptdf  5716  eusvobj2  5905  tfri3  6422  mapxpen  6906  fodjuomnilemdc  7205  cc3  7330  fimaxre2  11373  fprodcllemf  11759  fprodap0f  11782  fprodle  11786  zsupcllemstep  12085  bezoutlemmain  12138  bezoutlemzz  12142  exmidunben  12586  mulcncf  14787  limccnp2lem  14855