MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reu7 Structured version   Unicode version

Theorem reu7 3131
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 3126 . 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 1697 . . . . . . . 8  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
4 equcom 1693 . . . . . . . 8  |-  ( y  =  z  <->  z  =  y )
53, 4syl6bb 254 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  z  <->  z  =  y ) )
62, 5imbi12d 313 . . . . . 6  |-  ( x  =  y  ->  (
( ph  ->  x  =  z )  <->  ( ps  ->  z  =  y ) ) )
76cbvralv 2934 . . . . 5  |-  ( A. x  e.  A  ( ph  ->  x  =  z )  <->  A. y  e.  A  ( ps  ->  z  =  y ) )
87rexbii 2732 . . . 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 1697 . . . . . . 7  |-  ( z  =  x  ->  (
z  =  y  <->  x  =  y ) )
109imbi2d 309 . . . . . 6  |-  ( z  =  x  ->  (
( ps  ->  z  =  y )  <->  ( ps  ->  x  =  y ) ) )
1110ralbidv 2727 . . . . 5  |-  ( z  =  x  ->  ( A. y  e.  A  ( ps  ->  z  =  y )  <->  A. y  e.  A  ( ps  ->  x  =  y ) ) )
1211cbvrexv 2935 . . . 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 242 . . 3  |-  ( E. z  e.  A  A. x  e.  A  ( ph  ->  x  =  z )  <->  E. x  e.  A  A. y  e.  A  ( ps  ->  x  =  y ) )
1413anbi2i 677 . 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 242 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 178    /\ wa 360   A.wral 2707   E.wrex 2708   E!wreu 2709
This theorem is referenced by:  cshwssizesame  28322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715
  Copyright terms: Public domain W3C validator