Theorem eusv2i 4863

Description: Two ways to express single-valuedness of a class expression A ( x ). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
eusv2i	|- ( E! y A. x y = A -> E! y E. x y = A )

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem eusv2i

Step	Hyp	Ref	Expression
1		nfeu1 2480	. . 3 |- F/ y E! y A. x y = A
2		nfcvd 2765	. . . . . 6 |- ( E! y A. x y = A -> F/_ x y )
3		eusvnf 4861	. . . . . 6 |- ( E! y A. x y = A -> F/_ x A )
4	2, 3	nfeqd 2772	. . . . 5 |- ( E! y A. x y = A -> F/ x y = A )
5	4	nfrd 1717	. . . 4 |- ( E! y A. x y = A -> ( E. x y = A -> A. x y = A ) )
6		19.2 1892	. . . 4 |- ( A. x y = A -> E. x y = A )
7	5, 6	impbid1 215	. . 3 |- ( E! y A. x y = A -> ( E. x y = A <-> A. x y = A ) )
8	1, 7	eubid 2488	. 2 |- ( E! y A. x y = A -> ( E! y E. x y = A <-> E! y A. x y = A ) )
9	8	ibir 257	1 |- ( E! y A. x y = A -> E! y E. x y = A )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   E.wex 1704   E!weu 2470

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  eusv2nf  4864