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Theorem r19.2m 3449
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1617). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
Assertion
Ref Expression
r19.2m  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem r19.2m
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2200 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
21cbvexv 1890 . . 3  |-  ( E. x  x  e.  A  <->  E. z  z  e.  A
)
3 eleq1w 2200 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
43cbvexv 1890 . . 3  |-  ( E. z  z  e.  A  <->  E. y  y  e.  A
)
52, 4bitri 183 . 2  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
6 df-ral 2421 . . . . 5  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
7 exintr 1613 . . . . 5  |-  ( A. x ( x  e.  A  ->  ph )  -> 
( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
86, 7sylbi 120 . . . 4  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x
( x  e.  A  /\  ph ) ) )
9 df-rex 2422 . . . 4  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
108, 9syl6ibr 161 . . 3  |-  ( A. x  e.  A  ph  ->  ( E. x  x  e.  A  ->  E. x  e.  A  ph ) )
1110impcom 124 . 2  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
125, 11sylanbr 283 1  |-  ( ( E. y  y  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 1329   E.wex 1468    e. wcel 1480   A.wral 2416   E.wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-clel 2135  df-ral 2421  df-rex 2422
This theorem is referenced by:  intssunim  3793  riinm  3885  iinexgm  4079  xpiindim  4676  cnviinm  5080  eusvobj2  5760  iinerm  6501  suplocexprlemml  7531  rexfiuz  10768  r19.2uz  10772  climuni  11069  cncnp2m  12410
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