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Theorem eusvobj1 5764
Description: Specify the same object in two ways when class  B ( y ) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypothesis
Ref Expression
eusvobj1.1  |-  B  e. 
_V
Assertion
Ref Expression
eusvobj1  |-  ( E! x E. y  e.  A  x  =  B  ->  ( iota x E. y  e.  A  x  =  B )  =  ( iota x A. y  e.  A  x  =  B )
)
Distinct variable groups:    x, y, A   
x, B
Allowed substitution hint:    B( y)

Proof of Theorem eusvobj1
StepHypRef Expression
1 nfeu1 2010 . . 3  |-  F/ x E! x E. y  e.  A  x  =  B
2 eusvobj1.1 . . . 4  |-  B  e. 
_V
32eusvobj2 5763 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )
)
41, 3alrimi 1502 . 2  |-  ( E! x E. y  e.  A  x  =  B  ->  A. x ( E. y  e.  A  x  =  B  <->  A. y  e.  A  x  =  B ) )
5 iotabi 5100 . 2  |-  ( A. x ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )  ->  ( iota x E. y  e.  A  x  =  B )  =  ( iota x A. y  e.  A  x  =  B ) )
64, 5syl 14 1  |-  ( E! x E. y  e.  A  x  =  B  ->  ( iota x E. y  e.  A  x  =  B )  =  ( iota x A. y  e.  A  x  =  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   E!weu 1999   A.wral 2416   E.wrex 2417   _Vcvv 2686   iotacio 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-sn 3533  df-uni 3740  df-iota 5091
This theorem is referenced by: (None)
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