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Theorem rspec3 2643
Description: Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
Hypothesis
Ref Expression
rspec3.1  |-  A. x  e.  A  A. y  e.  B  A. z  e.  C  ph
Assertion
Ref Expression
rspec3  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )

Proof of Theorem rspec3
StepHypRef Expression
1 rspec3.1 . . . 4  |-  A. x  e.  A  A. y  e.  B  A. z  e.  C  ph
21rspec2 2642 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  A. z  e.  C  ph )
32r19.21bi 2641 . 2  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C )  ->  ph )
433impa 1146 1  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ral 2548
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