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Theorem eusvobj1 6276
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 2288 . . 3  |-  F/ x E! x E. y  e.  A  x  =  B
2 eusvobj1.1 . . . 4  |-  B  e. 
_V
32eusvobj2 6275 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )
)
41, 3alrimi 1825 . 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 5558 . 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 16 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379    e. wcel 1767   E!weu 2275   A.wral 2814   E.wrex 2815   _Vcvv 3113   iotacio 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-nul 3786  df-sn 4028  df-uni 4246  df-iota 5549
This theorem is referenced by: (None)
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