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Theorem iota2 5570
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 3117 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 simpl 457 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  A  e.  _V )
3 simpr 461 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  E! x ph )
4 iota2.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
54adantl 466 . . 3  |-  ( ( ( A  e.  _V  /\  E! x ph )  /\  x  =  A
)  ->  ( ph  <->  ps ) )
6 nfv 1678 . . . 4  |-  F/ x  A  e.  _V
7 nfeu1 2283 . . . 4  |-  F/ x E! x ph
86, 7nfan 1870 . . 3  |-  F/ x
( A  e.  _V  /\  E! x ph )
9 nfvd 1679 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/ x ps )
10 nfcvd 2625 . . 3  |-  ( ( A  e.  _V  /\  E! x ph )  ->  F/_ x A )
112, 3, 5, 8, 9, 10iota2df 5568 . 2  |-  ( ( A  e.  _V  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
121, 11sylan 471 1  |-  ( ( A  e.  B  /\  E! x ph )  -> 
( ps  <->  ( iota x ph )  =  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E!weu 2270   _Vcvv 3108   iotacio 5542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-v 3110  df-sbc 3327  df-un 3476  df-sn 4023  df-pr 4025  df-uni 4241  df-iota 5544
This theorem is referenced by:  pczpre  14221  pcdiv  14226  rngurd  27429  unirep  29795  ellimciota  31113  bj-nuliota  33544
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