Theorem eusvobj1 5519

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

Step	Hyp	Ref	Expression
1		nfeu1 1952	. . 3 |- F/ x E! x E. y e. A x = B
2		eusvobj1.1	. . . 4 |- B e. _V
3	2	eusvobj2 5518	. . 3 |- ( E! x E. y e. A x = B -> ( E. y e. A x = B <-> A. y e. A x = B ) )
4	1, 3	alrimi 1455	. 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 4896	. 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 ) )
6	4, 5	syl 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 ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   E!weu 1941   A.wral 2348   E.wrex 2349   _Vcvv 2601   iotacio 4885

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-sn 3404  df-uni 3602  df-iota 4887

This theorem is referenced by: (None)