Theorem eusv2 4504

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 4503	. 2 |- ( E! y E. x y = A <-> F/_ x A )
3		eusvnfb 4501	. . 3 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
4	1, 3	mpbiran2 944	. 2 |- ( E! y A. x y = A <-> F/_ x A )
5	2, 4	bitr4i 187	1 |- ( E! y E. x y = A <-> E! y A. x y = A )

Syntax hints:   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1515   E!weu 2054   e. wcel 2176   F/_wnfc 2335   _Vcvv 2772

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851

This theorem is referenced by: (None)