Theorem eusv2 4207

Description: Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypothesis

Ref	Expression
eusv2.1	|- A e. _V

Assertion

Ref	Expression
eusv2	|- ( E! y E. x y = A <-> E! y A. x y = A )

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem eusv2

Step	Hyp	Ref	Expression
1		eusv2.1	. . 3 |- A e. _V
2	1	eusv2nf 4206	. 2 |- ( E! y E. x y = A <-> F/_ x A )
3		eusvnfb 4204	. . 3 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
4	1, 3	mpbiran2 882	. 2 |- ( E! y A. x y = A <-> F/_ x A )
5	2, 4	bitr4i 185	1 |- ( E! y E. x y = A <-> E! y A. x y = A )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   F/_wnfc 2206   _Vcvv 2601

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-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602

This theorem is referenced by: (None)