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Theorem rspec3 2808
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 2807 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  A. z  e.  C  ph )
32r19.21bi 2806 . 2  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  e.  C )  ->  ph )
433impa 1149 1  |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   A.wral 2707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-ex 1552  df-ral 2712
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