Theorem eusv2 4865

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 4864	. 2 |- ( E! y E. x y = A <-> F/_ x A )
3		eusvnfb 4862	. . 3 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
4	1, 3	mpbiran2 954	. 2 |- ( E! y A. x y = A <-> F/_ x A )
5	2, 4	bitr4i 267	1 |- ( E! y E. x y = A <-> E! y A. x y = A )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   F/_wnfc 2751   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by: (None)