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Theorem r19.2m 3500
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1631). 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 2231 . . . 4  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
21cbvexv 1911 . . 3  |-  ( E. x  x  e.  A  <->  E. z  z  e.  A
)
3 eleq1w 2231 . . . 4  |-  ( z  =  y  ->  (
z  e.  A  <->  y  e.  A ) )
43cbvexv 1911 . . 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 2453 . . . . 5  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
7 exintr 1627 . . . . 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 2454 . . . 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 1346   E.wex 1485    e. wcel 2141   A.wral 2448   E.wrex 2449
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-clel 2166  df-ral 2453  df-rex 2454
This theorem is referenced by:  intssunim  3851  riinm  3943  iinexgm  4138  xpiindim  4746  cnviinm  5150  eusvobj2  5837  iinerm  6582  suplocexprlemml  7667  rexfiuz  10942  r19.2uz  10946  climuni  11245  pc2dvds  12272  cncnp2m  12986
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