Theorem eunex 4627

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 1552	. . 3 |- F/ y ph
2	1	eu3 2102	. 2 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
3		dtruex 4625	. . . . 5 |- E. x -. x = y
4		nfa1 1565	. . . . . 6 |- F/ x A. x ( ph -> x = y )
5		sp 1535	. . . . . . 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 1636	. . . . 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 1622	. . 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 1371   = wceq 1373   E.wex 1516   E!weu 2055

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649

This theorem is referenced by: (None)