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Theorem r19.27av 2854
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 2797 . . 3  |-  ( ps 
->  A. x  e.  A  ps )
32anim2i 569 . 2  |-  ( ( A. x  e.  A  ph 
/\  ps )  ->  ( A. x  e.  A  ph 
/\  A. x  e.  A  ps ) )
4 r19.26 2848 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4sylibr 212 1  |-  ( ( A. x  e.  A  ph 
/\  ps )  ->  A. x  e.  A  ( ph  /\ 
ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1587  df-nf 1590  df-ral 2719
This theorem is referenced by:  r19.28av  2855  txlm  19220  tx1stc  19222  spanuni  24946
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