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Theorem eusvobj1 6354
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 2166 . . 3  |-  F/ x E! x E. y  e.  A  x  =  B
2 eusvobj1.1 . . . 4  |-  B  e. 
_V
32eusvobj2 6353 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )
)
41, 3alrimi 1757 . 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 5244 . 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 15 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 176   A.wal 1530    = wceq 1632    e. wcel 1696   E!weu 2156   A.wral 2556   E.wrex 2557   _Vcvv 2801   iotacio 5233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-nul 3469  df-sn 3659  df-uni 3844  df-iota 5235
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