Theorem ralimdaa 2563

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)

Hypotheses

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

Assertion

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

Proof of Theorem ralimdaa

Step	Hyp	Ref	Expression
1		ralimdaa.1	. . 3 |- F/ x ph
2		ralimdaa.2	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	2	ex 115	. . . 4 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
4	3	a2d 26	. . 3 |- ( ph -> ( ( x e. A -> ps ) -> ( x e. A -> ch ) ) )
5	1, 4	alimd 1535	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) -> A. x ( x e. A -> ch ) ) )
6		df-ral 2480	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7		df-ral 2480	. 2 |- ( A. x e. A ch <-> A. x ( x e. A -> ch ) )
8	5, 6, 7	3imtr4g 205	1 |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   F/wnf 1474   e. wcel 2167   A.wral 2475

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 1461  ax-gen 1463  ax-4 1524

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480

This theorem is referenced by:  ralimdva  2564  mkvprop  7224  isomninnlem  15674  ismkvnnlem  15696