Theorem ralimdaa 2386

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 108	. . . 4 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
4	3	a2d 23	. . 3 |- ( ph -> ( ( x e. A -> ps ) -> ( x e. A -> ch ) ) )
5	1, 4	alimd 1414	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) -> A. x ( x e. A -> ch ) ) )
6		df-ral 2311	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7		df-ral 2311	. 2 |- ( A. x e. A ch <-> A. x ( x e. A -> ch ) )
8	5, 6, 7	3imtr4g 194	1 |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 97   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:  ralimdva  2387