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Theorem reu7 3294
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 3289 . 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 1799 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
4 equcom 1795 . . . . . . . 8  |-  ( y  =  z  <->  z  =  y )
53, 4syl6bb 261 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  z  <->  z  =  y ) )
62, 5imbi12d 320 . . . . . 6  |-  ( x  =  y  ->  (
( ph  ->  x  =  z )  <->  ( ps  ->  z  =  y ) ) )
76cbvralv 3084 . . . . 5  |-  ( A. x  e.  A  ( ph  ->  x  =  z )  <->  A. y  e.  A  ( ps  ->  z  =  y ) )
87rexbii 2959 . . . 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 1799 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
109imbi2d 316 . . . . . 6  |-  ( z  =  x  ->  (
( ps  ->  z  =  y )  <->  ( ps  ->  x  =  y ) ) )
1110ralbidv 2896 . . . . 5  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  z  =  y )  <->  A. y  e.  A  ( ps  ->  x  =  y ) ) )
1211cbvrexv 3085 . . . 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 249 . . 3  |-  ( E. z  e.  A  A. x  e.  A  ( ph  ->  x  =  z )  <->  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) )
1413anbi2i 694 . 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 249 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wral 2807   E.wrex 2808   E!wreu 2809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815
This theorem is referenced by:  cshwrepswhash1  14598
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