Theorem axextnd 3737

Description: A version of the Axiom of Extensionality with no distinct variable conditions.

Assertion

Ref	Expression
axextnd	|- E.x((x e. y <-> x e. z) -> y = z)

Proof of Theorem axextnd

Step	Hyp	Ref	Expression
1		eq6 826	. . . . . . . 8 |- (-. A.x x = y -> A.x -. A.x x = y)
2		eq6 826	. . . . . . . 8 |- (-. A.x x = z -> A.x -. A.x x = z)
3	1, 2	hban 704	. . . . . . 7 |- ((-. A.x x = y /\ -. A.x x = z) -> A.x(-. A.x x = y /\ -. A.x x = z))
4		ddeel2 1004	. . . . . . . . 9 |- (-. A.x x = y -> (w e. y -> A.x w e. y))
5	4	adantr 306	. . . . . . . 8 |- ((-. A.x x = y /\ -. A.x x = z) -> (w e. y -> A.x w e. y))
6		ddeel2 1004	. . . . . . . . 9 |- (-. A.x x = z -> (w e. z -> A.x w e. z))
7	6	adantl 305	. . . . . . . 8 |- ((-. A.x x = y /\ -. A.x x = z) -> (w e. z -> A.x w e. z))
8	3, 5, 7	hbbid 789	. . . . . . 7 |- ((-. A.x x = y /\ -. A.x x = z) -> ((w e. y <-> w e. z) -> A.x(w e. y <-> w e. z)))
9		a13b 819	. . . . . . . . 9 |- (w = x -> (w e. y <-> x e. y))
10		a13b 819	. . . . . . . . 9 |- (w = x -> (w e. z <-> x e. z))
11	9, 10	bibi12d 477	. . . . . . . 8 |- (w = x -> ((w e. y <-> w e. z) <-> (x e. y <-> x e. z)))
12	11	a1i 7	. . . . . . 7 |- ((-. A.x x = y /\ -. A.x x = z) -> (w = x -> ((w e. y <-> w e. z) <-> (x e. y <-> x e. z))))
13	3, 8, 12	cbvald 977	. . . . . 6 |- ((-. A.x x = y /\ -. A.x x = z) -> (A.w(w e. y <-> w e. z) <-> A.x(x e. y <-> x e. z)))
14		zfext2 1087	. . . . . 6 |- (A.w(w e. y <-> w e. z) -> y = z)
15	13, 14	syl6bir 188	. . . . 5 |- ((-. A.x x = y /\ -. A.x x = z) -> (A.x(x e. y <-> x e. z) -> y = z))
16		19.8a 712	. . . . 5 |- (y = z -> E.x y = z)
17	15, 16	syl6 23	. . . 4 |- ((-. A.x x = y /\ -. A.x x = z) -> (A.x(x e. y <-> x e. z) -> E.x y = z))
18	17	exp 291	. . 3 |- (-. A.x x = y -> (-. A.x x = z -> (A.x(x e. y <-> x e. z) -> E.x y = z)))
19		a9e 809	. . . . 5 |- E.x x = z
20		ax-8 798	. . . . . . 7 |- (x = y -> (x = z -> y = z))
21	20	a4s 682	. . . . . 6 |- (A.x x = y -> (x = z -> y = z))
22	21	del42 841	. . . . 5 |- (A.x x = y -> (E.x x = z -> E.x y = z))
23	19, 22	mpi 44	. . . 4 |- (A.x x = y -> E.x y = z)
24	23	a1d 14	. . 3 |- (A.x x = y -> (A.x(x e. y <-> x e. z) -> E.x y = z))
25		a9e 809	. . . . 5 |- E.x x = y
26		ax-8 798	. . . . . . . 8 |- (x = z -> (x = y -> z = y))
27		eqcom 811	. . . . . . . 8 |- (z = y -> y = z)
28	26, 27	syl6 23	. . . . . . 7 |- (x = z -> (x = y -> y = z))
29	28	a4s 682	. . . . . 6 |- (A.x x = z -> (x = y -> y = z))
30	29	del42 841	. . . . 5 |- (A.x x = z -> (E.x x = y -> E.x y = z))
31	25, 30	mpi 44	. . . 4 |- (A.x x = z -> E.x y = z)
32	31	a1d 14	. . 3 |- (A.x x = z -> (A.x(x e. y <-> x e. z) -> E.x y = z))
33	18, 24, 32	pm2.61ii 113	. 2 |- (A.x(x e. y <-> x e. z) -> E.x y = z)
34	33	19.35ri 756	1 |- E.x((x e. y <-> x e. z) -> y = z)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803

This theorem is referenced by:  zfcndext 3759

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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