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Theorem reu7 3093
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 3088 . 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 1692 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
4 equcom 1688 . . . . . . . 8  |-  ( y  =  z  <->  z  =  y )
53, 4syl6bb 253 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  z  <->  z  =  y ) )
62, 5imbi12d 312 . . . . . 6  |-  ( x  =  y  ->  (
( ph  ->  x  =  z )  <->  ( ps  ->  z  =  y ) ) )
76cbvralv 2896 . . . . 5  |-  ( A. x  e.  A  ( ph  ->  x  =  z )  <->  A. y  e.  A  ( ps  ->  z  =  y ) )
87rexbii 2695 . . . 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 1692 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
109imbi2d 308 . . . . . 6  |-  ( z  =  x  ->  (
( ps  ->  z  =  y )  <->  ( ps  ->  x  =  y ) ) )
1110ralbidv 2690 . . . . 5  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  z  =  y )  <->  A. y  e.  A  ( ps  ->  x  =  y ) ) )
1211cbvrexv 2897 . . . 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 241 . . 3  |-  ( E. z  e.  A  A. x  e.  A  ( ph  ->  x  =  z )  <->  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) )
1413anbi2i 676 . 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 241 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    <-> wb 177    /\ wa 359   A.wral 2670   E.wrex 2671   E!wreu 2672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678
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