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Theorem r19.29r 2646
Description: Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
r19.29r  |-  ( ( E. x  e.  A  ph 
/\  A. x  e.  A  ps )  ->  E. x  e.  A  ( ph  /\ 
ps ) )

Proof of Theorem r19.29r
StepHypRef Expression
1 r19.29 2645 . 2  |-  ( ( A. x  e.  A  ps  /\  E. x  e.  A  ph )  ->  E. x  e.  A  ( ps  /\  ph )
)
2 ancom 439 . 2  |-  ( ( E. x  e.  A  ph 
/\  A. x  e.  A  ps )  <->  ( A. x  e.  A  ps  /\  E. x  e.  A  ph )
)
3 ancom 439 . . 3  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
43rexbii 2532 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  E. x  e.  A  ( ps  /\  ph )
)
51, 2, 43imtr4i 259 1  |-  ( ( E. x  e.  A  ph 
/\  A. x  e.  A  ps )  ->  E. x  e.  A  ( ph  /\ 
ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wral 2509   E.wrex 2510
This theorem is referenced by:  rlimuni  11901  rlimno1  12004  neindisj2  16692  lmss  16858  fclsbas  17548  isfcf  17561  metcnp3  17918  bndth  18288  ellimc3  19061  cmptdst  24734  cover2  25524  bnj517  27606
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-ral 2513  df-rex 2514
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