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Theorem eusvobj1 5621
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 1959 . . 3  |-  F/ x E! x E. y  e.  A  x  =  B
2 eusvobj1.1 . . . 4  |-  B  e. 
_V
32eusvobj2 5620 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )
)
41, 3alrimi 1460 . 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 4976 . 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 103   A.wal 1287    = wceq 1289    e. wcel 1438   E!weu 1948   A.wral 2359   E.wrex 2360   _Vcvv 2619   iotacio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-sn 3447  df-uni 3649  df-iota 4967
This theorem is referenced by: (None)
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