Theorem rgen3 1721

Description: Generalization rule for restricted quantification.

Hypothesis

Ref	Expression
rgen3.1	|- ((x e. A /\ y e. B /\ z e. C) -> ph)

Assertion

Ref	Expression
rgen3	|- A.x e. A A.y e. B A.z e. C ph

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

Proof of Theorem rgen3

Step	Hyp	Ref	Expression
1		rgen3.1	. . . 4 |- ((x e. A /\ y e. B /\ z e. C) -> ph)
2	1	3expa 832	. . 3 |- (((x e. A /\ y e. B) /\ z e. C) -> ph)
3	2	r19.21aiva 1711	. 2 |- ((x e. A /\ y e. B) -> A.z e. C ph)
4	3	rgen2 1720	1 |- A.x e. A A.y e. B A.z e. C ph

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   e. wcel 956  A.wral 1642

This theorem is referenced by:  itlso 2858  zorn 4777  retopbas 7605  isgrpi 7992  isgrp2i 8026  cnring 8114  ringsn 8115  lnocoi 8365  0lnfn 9848  lnopco 9866  1cat 10572

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 776  df-ral 1646

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