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Theorem r19.29r 2697
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 2696 . 2  |-  ( ( A. x  e.  A  ps  /\  E. x  e.  A  ph )  ->  E. x  e.  A  ( ps  /\  ph )
)
2 ancom 437 . 2  |-  ( ( E. x  e.  A  ph 
/\  A. x  e.  A  ps )  <->  ( A. x  e.  A  ps  /\  E. x  e.  A  ph )
)
3 ancom 437 . . 3  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
43rexbii 2581 . 2  |-  ( E. x  e.  A  (
ph  /\  ps )  <->  E. x  e.  A  ( ps  /\  ph )
)
51, 2, 43imtr4i 257 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 4    /\ wa 358   A.wral 2556   E.wrex 2557
This theorem is referenced by:  2reu5  2986  rlimuni  12040  rlimno1  12143  neindisj2  16876  lmss  17042  fclsbas  17732  isfcf  17745  metcnp3  18102  bndth  18472  ellimc3  19245  lmxrge0  23390  esumcst  23451  cmptdst  25671  cover2  26461  bnj517  29233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-ral 2561  df-rex 2562
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