Theorem ralrimivvva 2708

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
ralrimivvva.1	|- ((ph /\ (x e. A /\ y e. B /\ z e. C)) -> ps)

Assertion

Ref	Expression
ralrimivvva	|- (ph -> A.x e. A A.y e. B A.z e. C ps)

Distinct variable groups:   ph,x,y,z   y,A,z   z,B

Allowed substitution hints:   ps(x,y,z)   A(x)   B(x,y)   C(x,y,z)

Proof of Theorem ralrimivvva

Step	Hyp	Ref	Expression
1		ralrimivvva.1	. . . . . 6 |- ((ph /\ (x e. A /\ y e. B /\ z e. C)) -> ps)
2	1	3exp2 1169	. . . . 5 |- (ph -> (x e. A -> (y e. B -> (z e. C -> ps))))
3	2	imp41 576	. . . 4 |- ((((ph /\ x e. A) /\ y e. B) /\ z e. C) -> ps)
4	3	ralrimiva 2698	. . 3 |- (((ph /\ x e. A) /\ y e. B) -> A.z e. C ps)
5	4	ralrimiva 2698	. 2 |- ((ph /\ x e. A) -> A.y e. B A.z e. C ps)
6	5	ralrimiva 2698	1 |- (ph -> A.x e. A A.y e. B A.z e. C ps)

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934   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-3an 936  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  caovassg  5627  caovdig  5633  caovdirg  5634