Theorem eunex 4285

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 1421	. . 3 |- F/ y ph
2	1	eu3 1946	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
3		dtruex 4283	. . . . 5 |- E. x -. x = y
4		nfa1 1434	. . . . . 6 |- F/ x A. x ( ph -> x = y )
5		sp 1401	. . . . . . 7 |- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
6	5	con3d 561	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( -. x = y -> -. ph ) )
7	4, 6	eximd 1503	. . . . 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 1489	. . 3 |- ( E. y A. x ( ph -> x = y ) -> E. x -. ph )
10	9	adantl 262	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> E. x -. ph )
11	2, 10	sylbi 114	1 |- ( E! x ph -> E. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   E.wex 1381   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by: (None)