Theorem axextnd 4931

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		hbnae 1147	. . . . . . . 8 |- (-. A.x x = y -> A.x -. A.x x = y)
2		hbnae 1147	. . . . . . . 8 |- (-. A.x x = z -> A.x -. A.x x = z)
3	1, 2	hban 1009	. . . . . . 7 |- ((-. A.x x = y /\ -. A.x x = z) -> A.x(-. A.x x = y /\ -. A.x x = z))
4		dveel2 1357	. . . . . . . . 9 |- (-. A.x x = y -> (w e. y -> A.x w e. y))
5	4	adantr 389	. . . . . . . 8 |- ((-. A.x x = y /\ -. A.x x = z) -> (w e. y -> A.x w e. y))
6		dveel2 1357	. . . . . . . . 9 |- (-. A.x x = z -> (w e. z -> A.x w e. z))
7	6	adantl 388	. . . . . . . 8 |- ((-. A.x x = y /\ -. A.x x = z) -> (w e. z -> A.x w e. z))
8	3, 5, 7	hbbid 1112	. . . . . . 7 |- ((-. A.x x = y /\ -. A.x x = z) -> ((w e. y <-> w e. z) -> A.x(w e. y <-> w e. z)))
9		elequ1 1136	. . . . . . . . 9 |- (w = x -> (w e. y <-> x e. y))
10		elequ1 1136	. . . . . . . . 9 |- (w = x -> (w e. z <-> x e. z))
11	9, 10	bibi12d 629	. . . . . . . 8 |- (w = x -> ((w e. y <-> w e. z) <-> (x e. y <-> x e. z)))
12	11	a1i 8	. . . . . . 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 1320	. . . . . 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 1461	. . . . . 6 |- (A.w(w e. y <-> w e. z) -> y = z)
15	13, 14	syl6bir 215	. . . . 5 |- ((-. A.x x = y /\ -. A.x x = z) -> (A.x(x e. y <-> x e. z) -> y = z))
16		19.8a 1029	. . . . 5 |- (y = z -> E.x y = z)
17	15, 16	syl6 22	. . . 4 |- ((-. A.x x = y /\ -. A.x x = z) -> (A.x(x e. y <-> x e. z) -> E.x y = z))
18	17	ex 373	. . 3 |- (-. A.x x = y -> (-. A.x x = z -> (A.x(x e. y <-> x e. z) -> E.x y = z)))
19		a9e 1125	. . . . 5 |- E.x x = z
20		hbae 1145	. . . . . 6 |- (A.x x = y -> A.xA.x x = y)
21		ax-8 964	. . . . . . 7 |- (x = y -> (x = z -> y = z))
22	21	a4s 984	. . . . . 6 |- (A.x x = y -> (x = z -> y = z))
23	20, 22	19.22d 1062	. . . . 5 |- (A.x x = y -> (E.x x = z -> E.x y = z))
24	19, 23	mpi 44	. . . 4 |- (A.x x = y -> E.x y = z)
25	24	a1d 12	. . 3 |- (A.x x = y -> (A.x(x e. y <-> x e. z) -> E.x y = z))
26		a9e 1125	. . . . 5 |- E.x x = y
27		hbae 1145	. . . . . 6 |- (A.x x = z -> A.xA.x x = z)
28		ax-8 964	. . . . . . . 8 |- (x = z -> (x = y -> z = y))
29		equcomi 1128	. . . . . . . 8 |- (z = y -> y = z)
30	28, 29	syl6 22	. . . . . . 7 |- (x = z -> (x = y -> y = z))
31	30	a4s 984	. . . . . 6 |- (A.x x = z -> (x = y -> y = z))
32	27, 31	19.22d 1062	. . . . 5 |- (A.x x = z -> (E.x x = y -> E.x y = z))
33	26, 32	mpi 44	. . . 4 |- (A.x x = z -> E.x y = z)
34	33	a1d 12	. . 3 |- (A.x x = z -> (A.x(x e. y <-> x e. z) -> E.x y = z))
35	18, 25, 34	pm2.61ii 130	. 2 |- (A.x(x e. y <-> x e. z) -> E.x y = z)
36	35	19.35ri 1077	1 |- E.x((x e. y <-> x e. z) -> y = z)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  zfcndext 4953

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

Copyright terms: Public domain