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Theorem eusvobj1 6264
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 2296 . . 3  |-  F/ x E! x E. y  e.  A  x  =  B
2 eusvobj1.1 . . . 4  |-  B  e. 
_V
32eusvobj2 6263 . . 3  |-  ( E! x E. y  e.  A  x  =  B  ->  ( E. y  e.  A  x  =  B 
<-> 
A. y  e.  A  x  =  B )
)
41, 3alrimi 1882 . 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 5543 . 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 1396    = wceq 1398    e. wcel 1823   E!weu 2284   A.wral 2804   E.wrex 2805   _Vcvv 3106   iotacio 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-nul 3784  df-sn 4017  df-uni 4236  df-iota 5534
This theorem is referenced by: (None)
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