Theorem rgen3 2449

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 1139	. . 3 |- ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ph )
3	2	ralrimiva 2435	. 2 |- ( ( x e. A /\ y e. B ) -> A. z e. C ph )
4	3	rgen2 2448	1 |- A. x e. A A. y e. B A. z e. C ph

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   e. wcel 1434   A.wral 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460

This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-ral 2354

This theorem is referenced by:  reg3exmidlemwe  4329  ltsopr  6848  ltsosr  7003  ltso  7256  xrltso  8947