Theorem rgen3 2637

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 1234	. . 3 |- ( ( ( x e. A /\ y e. B ) /\ z e. C ) -> ph )
3	2	ralrimiva 2623	. 2 |- ( ( x e. A /\ y e. B ) -> A. z e. C ph )
4	3	rgen2 2636	1 |- A. x e. A A. y e. B A. z e. C ph

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   e. wcel 2209   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This proof depends on definitions:  df-bi 117  df-3an 1011  df-nf 1514  df-ral 2533

This theorem is used by:  reg3exmidlemwe  4726  ltsopr  7963  ltsosr  8131  ltso  8403  aptap  8978  xrltso  10198  addcncntoplem  15662