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Theorem rgen3 2460
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
StepHypRef Expression
1 rgen3.1 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
213expa 1143 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C )  ->  ph )
32ralrimiva 2446 . 2  |-  ( ( x  e.  A  /\  y  e.  B )  ->  A. z  e.  C  ph )
43rgen2 2459 1  |-  A. x  e.  A  A. y  e.  B  A. z  e.  C  ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    e. wcel 1438   A.wral 2359
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 1381  ax-gen 1383  ax-4 1445  ax-17 1464
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-ral 2364
This theorem is referenced by:  reg3exmidlemwe  4392  ltsopr  7145  ltsosr  7300  ltso  7553  xrltso  9256
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