Theorem ralimdaa 2692

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/xph
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/xph
2		ralimdaa.2	. . . . 5 |- ((ph /\ x e. A) -> (ps -> ch))
3	2	ex 423	. . . 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 1764	. 2 |- (ph -> (A.x(x e. A -> ps) -> A.x(x e. A -> ch)))
6		df-ral 2620	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7		df-ral 2620	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
8	5, 6, 7	3imtr4g 261	1 |- (ph -> (A.x e. A ps -> A.x e. A ch))

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   F/wnf 1544   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  ralimdva  2693