Theorem eusv2 4378

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 4377	. 2 |- ( E! y E. x y = A <-> F/_ x A )
3		eusvnfb 4375	. . 3 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
4	1, 3	mpbiran2 925	. 2 |- ( E! y A. x y = A <-> F/_ x A )
5	2, 4	bitr4i 186	1 |- ( E! y E. x y = A <-> E! y A. x y = A )

Syntax hints:   <-> wb 104   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   E!weu 1999   F/_wnfc 2268   _Vcvv 2686

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737

This theorem is referenced by: (None)