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Theorem r19.28av 2682
Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.28av  |-  ( (
ph  /\  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  /\  ps )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.28av
StepHypRef Expression
1 r19.27av 2681 . 2  |-  ( ( A. x  e.  A  ps  /\  ph )  ->  A. x  e.  A  ( ps  /\  ph )
)
2 ancom 437 . 2  |-  ( (
ph  /\  A. x  e.  A  ps )  <->  ( A. x  e.  A  ps  /\  ph ) )
3 ancom 437 . . 3  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
43ralbii 2567 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  A. x  e.  A  ( ps  /\ 
ph ) )
51, 2, 43imtr4i 257 1  |-  ( (
ph  /\  A. x  e.  A  ps )  ->  A. x  e.  A  ( ph  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wral 2543
This theorem is referenced by:  rr19.28v  2910  fununi  5316  txlm  17342
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-tru 1310  df-nf 1532  df-ral 2548
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