Theorem eusvobj1 5944

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 2066	. . 3 |- F/ x E! x E. y e. A x = B
2		eusvobj1.1	. . . 4 |- B e. _V
3	2	eusvobj2 5943	. . 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 1546	. 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 5250	. 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 1371   = wceq 1373   E!weu 2055   e. wcel 2177   A.wral 2485   E.wrex 2486   _Vcvv 2773   iotacio 5239

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-csb 3098  df-sn 3644  df-uni 3857  df-iota 5241

This theorem is referenced by: (None)