Theorem eusvobj1 5754

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 2008	. . 3 |- F/ x E! x E. y e. A x = B
2		eusvobj1.1	. . . 4 |- B e. _V
3	2	eusvobj2 5753	. . 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 1502	. 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 5092	. 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 104   A.wal 1329   = wceq 1331   e. wcel 1480   E!weu 1997   A.wral 2414   E.wrex 2415   _Vcvv 2681   iotacio 5081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-sn 3528  df-uni 3732  df-iota 5083

This theorem is referenced by: (None)