Theorem eunex 4561

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 1528	. . 3 |- F/ y ph
2	1	eu3 2072	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
3		dtruex 4559	. . . . 5 |- E. x -. x = y
4		nfa1 1541	. . . . . 6 |- F/ x A. x ( ph -> x = y )
5		sp 1511	. . . . . . 7 |- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
6	5	con3d 631	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( -. x = y -> -. ph ) )
7	4, 6	eximd 1612	. . . . 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 1598	. . 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 1351   = wceq 1353   E.wex 1492   E!weu 2026

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599

This theorem is referenced by: (None)