Theorem rgen3 1724

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 833	. . 3 |- (((x e. A /\ y e. B) /\ z e. C) -> ph)
3	2	r19.21aiva 1714	. 2 |- ((x e. A /\ y e. B) -> A.z e. C ph)
4	3	rgen2 1723	1 |- A.x e. A A.y e. B A.z e. C ph

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   e. wcel 958  A.wral 1645

This theorem is referenced by:  itlso 2863  zorn 4797  retopbas 7655  isgrpi 8042  isgrp2i 8076  cnring 8162  ringsn 8163  lnocoi 8418  0lnfn 9909  lnopco 9928  1cat 10692

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ral 1649

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