Theorem rgen3 2557

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 1198	. . 3 |- ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ph )
3	2	ralrimiva 2543	. 2 |- ( ( x e. A /\ y e. B ) -> A. z e. C ph )
4	3	rgen2 2556	1 |- A. x e. A A. y e. B A. z e. C ph

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   e. wcel 2141   A.wral 2448

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 1440  ax-gen 1442  ax-4 1503  ax-17 1519

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-ral 2453

This theorem is referenced by:  reg3exmidlemwe  4563  ltsopr  7558  ltsosr  7726  ltso  7997  xrltso  9753  addcncntoplem  13345