Theorem eunex 3500

Description: Existential uniqueness implies there is a value for which the wff argument is false.

Assertion

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

Proof of Theorem eunex

Step	Hyp	Ref	Expression
1		dtru 3498	. . . . 5 |- -. A.x x = y
2		alim 1340	. . . . 5 |- (A.x(ph -> x = y) -> (A.xph -> A.x x = y))
3	1, 2	mtoi 122	. . . 4 |- (A.x(ph -> x = y) -> -. A.xph)
4	3	19.23aiv 1674	. . 3 |- (E.yA.x(ph -> x = y) -> -. A.xph)
5	4	adantl 424	. 2 |- ((E.xph /\ E.yA.x(ph -> x = y)) -> -. A.xph)
6		ax-17 1317	. . 3 |- (ph -> A.yph)
7	6	eu3 1792	. 2 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))
8		exnal 1385	. 2 |- (E.x -. ph <-> -. A.xph)
9	5, 7, 8	3imtr4i 236	1 |- (E!xph -> E.x -. ph)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  E!weu 1771

This theorem is referenced by:  eufromeq2 3829  unnt 14158  amosym1 14250

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-nul 3445  ax-pow 3481

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775

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