Theorem eunex 4578

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 1539	. . 3 |- F/ y ph
2	1	eu3 2084	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
3		dtruex 4576	. . . . 5 |- E. x -. x = y
4		nfa1 1552	. . . . . 6 |- F/ x A. x ( ph -> x = y )
5		sp 1522	. . . . . . 7 |- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
6	5	con3d 632	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( -. x = y -> -. ph ) )
7	4, 6	eximd 1623	. . . . 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 1609	. . 3 |- ( E. y A. x ( ph -> x = y ) -> E. x -. ph )
10	9	adantl 277	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> E. x -. ph )
11	2, 10	sylbi 121	1 |- ( E! x ph -> E. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   E.wex 1503   E!weu 2038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613

This theorem is referenced by: (None)