Theorem dtruALT 2755

Description: A version of dtru 2779 ("two things exist") proved directly from the axioms (no set theory definitions).

Assertion

Ref	Expression
dtruALT	|- -. A.x x = y

Distinct variable group:   x,y

Proof of Theorem dtruALT

Step	Hyp	Ref	Expression
1		eeanv 1325	. . . . 5 |- (E.wE.z(x e. w /\ -. x e. z) <-> (E.w x e. w /\ E.z -. x e. z))
2		ax-pow 2749	. . . . . 6 |- E.wA.z(A.y(y e. z -> y e. x) -> z e. w)
3		id 59	. . . . . . . . 9 |- (y e. x -> y e. x)
4	3	ax-gen 965	. . . . . . . 8 |- A.y(y e. x -> y e. x)
5		elequ2 1139	. . . . . . . . . . . 12 |- (z = x -> (y e. z <-> y e. x))
6	5	imbi1d 615	. . . . . . . . . . 11 |- (z = x -> ((y e. z -> y e. x) <-> (y e. x -> y e. x)))
7	6	albidv 1280	. . . . . . . . . 10 |- (z = x -> (A.y(y e. z -> y e. x) <-> A.y(y e. x -> y e. x)))
8		elequ1 1138	. . . . . . . . . 10 |- (z = x -> (z e. w <-> x e. w))
9	7, 8	imbi12d 628	. . . . . . . . 9 |- (z = x -> ((A.y(y e. z -> y e. x) -> z e. w) <-> (A.y(y e. x -> y e. x) -> x e. w)))
10	9	a4v 1274	. . . . . . . 8 |- (A.z(A.y(y e. z -> y e. x) -> z e. w) -> (A.y(y e. x -> y e. x) -> x e. w))
11	4, 10	mpi 44	. . . . . . 7 |- (A.z(A.y(y e. z -> y e. x) -> z e. w) -> x e. w)
12	11	19.22i 1042	. . . . . 6 |- (E.wA.z(A.y(y e. z -> y e. x) -> z e. w) -> E.w x e. w)
13	2, 12	ax-mp 7	. . . . 5 |- E.w x e. w
14		ax-nul 2716	. . . . . 6 |- E.zA.x -. x e. z
15		ax-4 975	. . . . . . 7 |- (A.x -. x e. z -> -. x e. z)
16	15	19.22i 1042	. . . . . 6 |- (E.zA.x -. x e. z -> E.z -. x e. z)
17	14, 16	ax-mp 7	. . . . 5 |- E.z -. x e. z
18	1, 13, 17	mpbir2an 732	. . . 4 |- E.wE.z(x e. w /\ -. x e. z)
19		ax-14 972	. . . . . . . 8 |- (w = z -> (x e. w -> x e. z))
20	19	com12 11	. . . . . . 7 |- (x e. w -> (w = z -> x e. z))
21	20	con3d 95	. . . . . 6 |- (x e. w -> (-. x e. z -> -. w = z))
22	21	imp 350	. . . . 5 |- ((x e. w /\ -. x e. z) -> -. w = z)
23	22	19.22i2 1043	. . . 4 |- (E.wE.z(x e. w /\ -. x e. z) -> E.wE.z -. w = z)
24	18, 23	ax-mp 7	. . 3 |- E.wE.z -. w = z
25		equequ2 1137	. . . . . . 7 |- (z = y -> (w = z <-> w = y))
26	25	negbid 613	. . . . . 6 |- (z = y -> (-. w = z <-> -. w = y))
27		ax-8 966	. . . . . . . 8 |- (x = w -> (x = y -> w = y))
28	27	con3d 95	. . . . . . 7 |- (x = w -> (-. w = y -> -. x = y))
29	28	a4imev 1275	. . . . . 6 |- (-. w = y -> E.x -. x = y)
30	26, 29	syl6bi 214	. . . . 5 |- (z = y -> (-. w = z -> E.x -. x = y))
31		ax-8 966	. . . . . . . 8 |- (x = z -> (x = y -> z = y))
32	31	con3d 95	. . . . . . 7 |- (x = z -> (-. z = y -> -. x = y))
33	32	a4imev 1275	. . . . . 6 |- (-. z = y -> E.x -. x = y)
34	33	a1d 12	. . . . 5 |- (-. z = y -> (-. w = z -> E.x -. x = y))
35	30, 34	pm2.61i 126	. . . 4 |- (-. w = z -> E.x -. x = y)
36	35	19.23aivv 1298	. . 3 |- (E.wE.z -. w = z -> E.x -. x = y)
37	24, 36	ax-mp 7	. 2 |- E.x -. x = y
38		exnal 1040	. 2 |- (E.x -. x = y <-> -. A.x x = y)
39	37, 38	mpbi 189	1 |- -. A.x x = y

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982

This theorem is referenced by:  ax16b 2756

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-nul 2716  ax-pow 2749

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983

Copyright terms: Public domain