Theorem eunex 4484

Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
eunex	|- ( E! x ph -> E. x -. ph )

Proof of Theorem eunex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1509	. . 3 |- F/ y ph
2	1	eu3 2046	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
3		dtruex 4482	. . . . 5 |- E. x -. x = y
4		nfa1 1522	. . . . . 6 |- F/ x A. x ( ph -> x = y )
5		sp 1489	. . . . . . 7 |- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
6	5	con3d 621	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( -. x = y -> -. ph ) )
7	4, 6	eximd 1592	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( E. x -. x = y -> E. x -. ph ) )
8	3, 7	mpi 15	. . . 4 |- ( A. x ( ph -> x = y ) -> E. x -. ph )
9	8	exlimiv 1578	. . 3 |- ( E. y A. x ( ph -> x = y ) -> E. x -. ph )
10	9	adantl 275	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> E. x -. ph )
11	2, 10	sylbi 120	1 |- ( E! x ph -> E. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1330   = wceq 1332   E.wex 1469   E!weu 2000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538

This theorem is referenced by: (None)