Theorem ralrimi 2390

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 1415	. 2 |- ( ph -> A. x ( x e. A -> ps ) )
4		df-ral 2311	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
5	3, 4	sylibr 137	1 |- ( ph -> A. x e. A ps )

Syntax hints:   -> wi 4   A.wal 1241   F/wnf 1349   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by:  ralrimiv  2391  reximdai  2417  r19.12  2422  rexlimd  2430  rexlimd2  2431  r19.29af2  2452  r19.37  2462  ralidm  3321  iineq2d  3677  mpteq2da  3846  onintonm  4243  mpteqb  5261  eusvobj2  5498  tfri3  5953