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Theorem r19.2m 3373
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1575). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
Assertion
Ref Expression
r19.2m  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem r19.2m
StepHypRef Expression
1 df-ral 2365 . . . 4  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
2 exintr 1571 . . . 4  |-  ( A. x ( x  e.  A  ->  ph )  -> 
( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
31, 2sylbi 120 . . 3  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
4 df-rex 2366 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
53, 4syl6ibr 161 . 2  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x  e.  A  ph ) )
65impcom 124 1  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1288   E.wex 1427    e. wcel 1439   A.wral 2360   E.wrex 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473
This theorem depends on definitions:  df-bi 116  df-ral 2365  df-rex 2366
This theorem is referenced by:  intssunim  3716  riinm  3808  trintssmOLD  3959  iinexgm  3996  xpiindim  4586  cnviinm  4985  eusvobj2  5652  iinerm  6378  rexfiuz  10483  r19.2uz  10487  climuni  10742
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