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Theorem r19.27av 2592
Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.27av  |-  ( ( A. x  e.  A  ph 
/\  ps )  ->  A. x  e.  A  ( ph  /\ 
ps ) )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem r19.27av
StepHypRef Expression
1 ax-1 6 . . . 4  |-  ( ps 
->  ( x  e.  A  ->  ps ) )
21ralrimiv 2529 . . 3  |-  ( ps 
->  A. x  e.  A  ps )
32anim2i 340 . 2  |-  ( ( A. x  e.  A  ph 
/\  ps )  ->  ( A. x  e.  A  ph 
/\  A. x  e.  A  ps ) )
4 r19.26 2583 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4sylibr 133 1  |-  ( ( A. x  e.  A  ph 
/\  ps )  ->  A. x  e.  A  ( ph  /\ 
ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 2128   A.wral 2435
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 1427  ax-gen 1429  ax-4 1490  ax-17 1506
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440
This theorem is referenced by:  r19.28av  2593  fimaxre2  11126
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