Theorem eusvobj1 5801

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 2014	. . 3 |- F/ x E! x E. y e. A x = B
2		eusvobj1.1	. . . 4 |- B e. _V
3	2	eusvobj2 5800	. . 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 1499	. 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 5137	. 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 1330   = wceq 1332   E!weu 2003   e. wcel 2125   A.wral 2432   E.wrex 2433   _Vcvv 2709   iotacio 5126

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-csb 3028  df-sn 3562  df-uni 3769  df-iota 5128

This theorem is referenced by: (None)