Theorem bj-dtru 31163

Description: Remove dependency on ax-13 2055 from dtru 4616. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-dtru	|- -. A. x x = y

Distinct variable group:   x, y

Proof of Theorem bj-dtru

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-el 31162	. . . . 5 |- E. w x e. w
2		ax-nul 4556	. . . . . 6 |- E. z A. x -. x e. z
3		sp 1912	. . . . . 6 |- ( A. x -. x e. z -> -. x e. z )
4	2, 3	eximii 1705	. . . . 5 |- E. z -. x e. z
5		eeanv 2045	. . . . 5 |- ( E. w E. z ( x e. w /\ -. x e. z ) <-> ( E. w x e. w /\ E. z -. x e. z ) )
6	1, 4, 5	mpbir2an 928	. . . 4 |- E. w E. z ( x e. w /\ -. x e. z )
7		ax-9 1874	. . . . . . 7 |- ( w = z -> ( x e. w -> x e. z ) )
8	7	com12 32	. . . . . 6 |- ( x e. w -> ( w = z -> x e. z ) )
9	8	con3dimp 442	. . . . 5 |- ( ( x e. w /\ -. x e. z ) -> -. w = z )
10	9	2eximi 1704	. . . 4 |- ( E. w E. z ( x e. w /\ -. x e. z ) -> E. w E. z -. w = z )
11	6, 10	ax-mp 5	. . 3 |- E. w E. z -. w = z
12		equequ2 1851	. . . . . . 7 |- ( z = y -> ( w = z <-> w = y ) )
13	12	notbid 295	. . . . . 6 |- ( z = y -> ( -. w = z <-> -. w = y ) )
14		ax-7 1841	. . . . . . . 8 |- ( x = w -> ( x = y -> w = y ) )
15	14	con3d 138	. . . . . . 7 |- ( x = w -> ( -. w = y -> -. x = y ) )
16	15	bj-spimevv 31063	. . . . . 6 |- ( -. w = y -> E. x -. x = y )
17	13, 16	syl6bi 231	. . . . 5 |- ( z = y -> ( -. w = z -> E. x -. x = y ) )
18		ax-7 1841	. . . . . . . 8 |- ( x = z -> ( x = y -> z = y ) )
19	18	con3d 138	. . . . . . 7 |- ( x = z -> ( -. z = y -> -. x = y ) )
20	19	bj-spimevv 31063	. . . . . 6 |- ( -. z = y -> E. x -. x = y )
21	20	a1d 26	. . . . 5 |- ( -. z = y -> ( -. w = z -> E. x -. x = y ) )
22	17, 21	pm2.61i 167	. . . 4 |- ( -. w = z -> E. x -. x = y )
23	22	exlimivv 1770	. . 3 |- ( E. w E. z -. w = z -> E. x -. x = y )
24	11, 23	ax-mp 5	. 2 |- E. x -. x = y
25		exnal 1695	. 2 |- ( E. x -. x = y <-> -. A. x x = y )
26	24, 25	mpbi 211	1 |- -. A. x x = y

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   A.wal 1435   E.wex 1659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-nul 4556  ax-pow 4603

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664

This theorem is referenced by:  bj-axc16b  31164  bj-eunex  31165  bj-dtrucor  31166  bj-dvdemo1  31168