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Theorem iota2 5591
Description: The unique element such that  ph. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)
Hypothesis
Ref Expression
iota2.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
iota2  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem iota2
StepHypRef Expression
1 elex 3089 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 simpl 458 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  A  e.  _V )
3 simpr 462 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  E! x ph )
4 iota2.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54adantl 467 . . 3  |-  ( ( ( A  e.  _V  /\  E! x ph )  /\  x  =  A
)  ->  ( ph  <->  ps ) )
6 nfv 1755 . . . 4  |-  F/ x  A  e.  _V
7 nfeu1 2279 . . . 4  |-  F/ x E! x ph
86, 7nfan 1988 . . 3  |-  F/ x
( A  e.  _V  /\  E! x ph )
9 nfvd 1756 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/ x ps )
10 nfcvd 2581 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/_ x A )
112, 3, 5, 8, 9, 10iota2df 5589 . 2  |-  ( ( A  e.  _V  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
121, 11sylan 473 1  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   E!weu 2269   _Vcvv 3080   iotacio 5563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-sbc 3300  df-un 3441  df-sn 3999  df-pr 4001  df-uni 4220  df-iota 5565
This theorem is referenced by:  pczpre  14796  pcdiv  14801  rngurd  28559  bj-nuliota  31590  unirep  32003  ellimciota  37634
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