Theorem ralrimivvva 2402

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	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. B /\ z e. C ) ) -> ps )
2	1	3anassrs 1126	. . . 4 |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ z e. C ) -> ps )
3	2	ralrimiva 2392	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> A. z e. C ps )
4	3	ralrimiva 2392	. 2 |- ( ( ph /\ x e. A ) -> A. y e. B A. z e. C ps )
5	4	ralrimiva 2392	1 |- ( ph -> A. x e. A A. y e. B A. z e. C ps )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   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  ax-17 1419

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-ral 2311

This theorem is referenced by:  ispod  4041  swopolem  4042  ordwe  4300  wessep  4302  isopolem  5461  caovassg  5659  caovcang  5662  caovordig  5666  caovordg  5668  caovdig  5675  caovdirg  5678  caoftrn  5736