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Theorem reu7 2810
Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)
Hypothesis
Ref Expression
rmo4.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
reu7  |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph 
/\  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) ) )
Distinct variable groups:    x, y, A    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem reu7
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 reu3 2805 . 2  |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph 
/\  E. z  e.  A  A. x  e.  A  ( ph  ->  x  =  z ) ) )
2 rmo4.1 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 equequ1 1645 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
4 equcom 1639 . . . . . . . 8  |-  ( y  =  z  <->  z  =  y )
53, 4syl6bb 194 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  z  <->  z  =  y ) )
62, 5imbi12d 232 . . . . . 6  |-  ( x  =  y  ->  (
( ph  ->  x  =  z )  <->  ( ps  ->  z  =  y ) ) )
76cbvralv 2590 . . . . 5  |-  ( A. x  e.  A  ( ph  ->  x  =  z )  <->  A. y  e.  A  ( ps  ->  z  =  y ) )
87rexbii 2385 . . . 4  |-  ( E. z  e.  A  A. x  e.  A  ( ph  ->  x  =  z )  <->  E. z  e.  A  A. y  e.  A  ( ps  ->  z  =  y ) )
9 equequ1 1645 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
109imbi2d 228 . . . . . 6  |-  ( z  =  x  ->  (
( ps  ->  z  =  y )  <->  ( ps  ->  x  =  y ) ) )
1110ralbidv 2380 . . . . 5  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  z  =  y )  <->  A. y  e.  A  ( ps  ->  x  =  y ) ) )
1211cbvrexv 2591 . . . 4  |-  ( E. z  e.  A  A. y  e.  A  ( ps  ->  z  =  y )  <->  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) )
138, 12bitri 182 . . 3  |-  ( E. z  e.  A  A. x  e.  A  ( ph  ->  x  =  z )  <->  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) )
1413anbi2i 445 . 2  |-  ( ( E. x  e.  A  ph 
/\  E. z  e.  A  A. x  e.  A  ( ph  ->  x  =  z ) )  <->  ( E. x  e.  A  ph  /\  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) ) )
151, 14bitri 182 1  |-  ( E! x  e.  A  ph  <->  ( E. x  e.  A  ph 
/\  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wral 2359   E.wrex 2360   E!wreu 2361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367
This theorem is referenced by: (None)
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