Theorem eusvobj1 6015

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 2090	. . 3 |- F/ x E! x E. y e. A x = B
2		eusvobj1.1	. . . 4 |- B e. _V
3	2	eusvobj2 6014	. . 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 1571	. 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 5303	. 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 105   A.wal 1396   = wceq 1398   E!weu 2079   e. wcel 2202   A.wral 2511   E.wrex 2512   _Vcvv 2803   iotacio 5291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-sn 3679  df-uni 3899  df-iota 5293

This theorem is referenced by: (None)