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Theorem euuni2OLD 26348
Description: The unique element such that  ph. (Moved to iota2 5245 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 5245 . 2  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
3 iotauni 5231 . . . 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 2291 . 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 1623    e. wcel 1684   E!weu 2143   {cab 2269   U.cuni 3827   iotacio 5217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219
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