Theorem ralrimivvva 2580

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 1231	. . . 4 |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ z e. C ) -> ps )
3	2	ralrimiva 2570	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> A. z e. C ps )
4	3	ralrimiva 2570	. 2 |- ( ( ph /\ x e. A ) -> A. y e. B A. z e. C ps )
5	4	ralrimiva 2570	1 |- ( ph -> A. x e. A A. y e. B A. z e. C ps )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   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  ax-17 1540

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-ral 2480

This theorem is referenced by:  ispod  4339  swopolem  4340  ordwe  4612  wessep  4614  isopolem  5869  caovassg  6082  caovcang  6085  caovordig  6089  caovordg  6091  caovdig  6098  caovdirg  6101  caoftrn  6163  netap  7321  2omotaplemap  7324  isrngd  13509  isringd  13597  aprap  13842  islmodd  13849  rnglidlmsgrp  14053  rnglidlrng  14054