Theorem ralrimivvva 2540

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 1211	. . . 4 |- ( ( ( ( ph /\ x e. A ) /\ y e. B ) /\ z e. C ) -> ps )
3	2	ralrimiva 2530	. . 3 |- ( ( ( ph /\ x e. A ) /\ y e. B ) -> A. z e. C ps )
4	3	ralrimiva 2530	. 2 |- ( ( ph /\ x e. A ) -> A. y e. B A. z e. C ps )
5	4	ralrimiva 2530	1 |- ( ph -> A. x e. A A. y e. B A. z e. C ps )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   e. wcel 2128   A.wral 2435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1441  df-ral 2440

This theorem is referenced by:  ispod  4266  swopolem  4267  ordwe  4537  wessep  4539  isopolem  5774  caovassg  5981  caovcang  5984  caovordig  5988  caovordg  5990  caovdig  5997  caovdirg  6000  caoftrn  6059