Theorem ralrimdv 2549

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)

Hypothesis

Ref	Expression
ralrimdv.1	|- ( ph -> ( ps -> ( x e. A -> ch ) ) )

Assertion

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

Distinct variable groups:   ph, x   ps, x

Allowed substitution hints:   ch( x)   A( x)

Proof of Theorem ralrimdv

Step	Hyp	Ref	Expression
1		nfv 1521	. 2 |- F/ x ph
2		nfv 1521	. 2 |- F/ x ps
3		ralrimdv.1	. 2 |- ( ph -> ( ps -> ( x e. A -> ch ) ) )
4	1, 2, 3	ralrimd 2548	1 |- ( ph -> ( ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   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  ax-17 1519

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

This theorem is referenced by:  ralrimdva  2550  ralrimivv  2551  nneneq  6835  fzrevral  10061  topbas  12861  neipsm  12948  cnpnei  13013  metcnp3  13305