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Theorem euuni2OLD 26451
Description: The unique element such that  ph. (Moved to iota2 5261 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Oct-2012.) (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
euuni2OLD.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
euuni2OLD  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  U. { x  |  ph }  =  A ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem euuni2OLD
StepHypRef Expression
1 euuni2OLD.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21iota2 5261 . 2  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
3 iotauni 5247 . . . 4  |-  ( E! x ph  ->  ( iota x ph )  = 
U. { x  | 
ph } )
43adantl 452 . . 3  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( iota x ph )  =  U. { x  | 
ph } )
54eqeq1d 2304 . 2  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ( iota x ph )  =  A  <->  U. { x  |  ph }  =  A )
)
62, 5bitrd 244 1  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  U. { x  |  ph }  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282   U.cuni 3843   iotacio 5233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235
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