Theorem eusv2 4469

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 4468	. 2 |- ( E! y E. x y = A <-> F/_ x A )
3		eusvnfb 4466	. . 3 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
4	1, 3	mpbiran2 942	. 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 1361   = wceq 1363   E.wex 1502   E!weu 2036   e. wcel 2158   F/_wnfc 2316   _Vcvv 2749

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-sn 3610  df-pr 3611  df-uni 3822

This theorem is referenced by: (None)