Theorem eusv2nf 4864

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

Hypothesis

Ref	Expression
eusv2.1	|- A e. _V

Assertion

Ref	Expression
eusv2nf	|- ( E! y E. x y = A <-> F/_ x A )

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem eusv2nf

Step	Hyp	Ref	Expression
1		nfeu1 2480	. . . 4 |- F/ y E! y E. x y = A
2		nfe1 2027	. . . . . . 7 |- F/ x E. x y = A
3	2	nfeu 2486	. . . . . 6 |- F/ x E! y E. x y = A
4		eusv2.1	. . . . . . . . 9 |- A e. _V
5	4	isseti 3209	. . . . . . . 8 |- E. y y = A
6		19.8a 2052	. . . . . . . . 9 |- ( y = A -> E. x y = A )
7	6	ancri 575	. . . . . . . 8 |- ( y = A -> ( E. x y = A /\ y = A ) )
8	5, 7	eximii 1764	. . . . . . 7 |- E. y ( E. x y = A /\ y = A )
9		eupick 2536	. . . . . . 7 |- ( ( E! y E. x y = A /\ E. y ( E. x y = A /\ y = A ) ) -> ( E. x y = A -> y = A ) )
10	8, 9	mpan2 707	. . . . . 6 |- ( E! y E. x y = A -> ( E. x y = A -> y = A ) )
11	3, 10	alrimi 2082	. . . . 5 |- ( E! y E. x y = A -> A. x ( E. x y = A -> y = A ) )
12		nf6 2117	. . . . 5 |- ( F/ x y = A <-> A. x ( E. x y = A -> y = A ) )
13	11, 12	sylibr 224	. . . 4 |- ( E! y E. x y = A -> F/ x y = A )
14	1, 13	alrimi 2082	. . 3 |- ( E! y E. x y = A -> A. y F/ x y = A )
15		dfnfc2 4454	. . . 4 |- ( A. x A e. _V -> ( F/_ x A <-> A. y F/ x y = A ) )
16	15, 4	mpg 1724	. . 3 |- ( F/_ x A <-> A. y F/ x y = A )
17	14, 16	sylibr 224	. 2 |- ( E! y E. x y = A -> F/_ x A )
18		eusvnfb 4862	. . . 4 |- ( E! y A. x y = A <-> ( F/_ x A /\ A e. _V ) )
19	4, 18	mpbiran2 954	. . 3 |- ( E! y A. x y = A <-> F/_ x A )
20		eusv2i 4863	. . 3 |- ( E! y A. x y = A -> E! y E. x y = A )
21	19, 20	sylbir 225	. 2 |- ( F/_ x A -> E! y E. x y = A )
22	17, 21	impbii 199	1 |- ( E! y E. x y = A <-> F/_ x A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   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:  eusv2  4865