Theorem rgen3 2712

Description: Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)

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

Allowed substitution hints:   ph(x, y, z)   A(x)   B(x, y)   C(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 1151	. . 3 |- (((x e. A /\ y e. B) /\ z e. C) -> ph)
3	2	ralrimiva 2698	. 2 |- ((x e. A /\ y e. B) -> A.z e. C ph)
4	3	rgen2 2711	1 |- A.x e. A A.y e. B A.z e. C ph

This proof depends on syntax axioms:   -> wi 4   /\ wa 358   /\ w3a 934   e. wcel 1710  A.wral 2615

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is used by: (None)