Theorem ax11eq 1402

Description: Basis step for constructing a substitution instance of ax-11o 1255 without using ax-11o 1255. Atomic formula for equality predicate.

Assertion

Ref	Expression
ax11eq	|- (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w))))

Proof of Theorem ax11eq

Step	Hyp	Ref	Expression
1		19.26 1103	. . 3 |- (A.x(x = z /\ x = w) <-> (A.x x = z /\ A.x x = w))
2		equid 1162	. . . . . . . 8 |- x = x
3	2	a1i 8	. . . . . . 7 |- (x = y -> x = x)
4	3	ax-gen 999	. . . . . 6 |- A.x(x = y -> x = x)
5	4	a1i 8	. . . . 5 |- (x = x -> A.x(x = y -> x = x))
6		equequ1 1171	. . . . . . . . 9 |- (x = z -> (x = x <-> z = x))
7		equequ2 1172	. . . . . . . . 9 |- (x = w -> (z = x <-> z = w))
8	6, 7	sylan9bb 543	. . . . . . . 8 |- ((x = z /\ x = w) -> (x = x <-> z = w))
9	8	a4s 1020	. . . . . . 7 |- (A.x(x = z /\ x = w) -> (x = x <-> z = w))
10		hba1 1039	. . . . . . . 8 |- (A.x(x = z /\ x = w) -> A.xA.x(x = z /\ x = w))
11	9	imbi2d 615	. . . . . . . 8 |- (A.x(x = z /\ x = w) -> ((x = y -> x = x) <-> (x = y -> z = w)))
12	10, 11	albid 1140	. . . . . . 7 |- (A.x(x = z /\ x = w) -> (A.x(x = y -> x = x) <-> A.x(x = y -> z = w)))
13	9, 12	imbi12d 629	. . . . . 6 |- (A.x(x = z /\ x = w) -> ((x = x -> A.x(x = y -> x = x)) <-> (z = w -> A.x(x = y -> z = w))))
14	13	adantr 389	. . . . 5 |- ((A.x(x = z /\ x = w) /\ (-. A.x x = y /\ x = y)) -> ((x = x -> A.x(x = y -> x = x)) <-> (z = w -> A.x(x = y -> z = w))))
15	5, 14	mpbii 191	. . . 4 |- ((A.x(x = z /\ x = w) /\ (-. A.x x = y /\ x = y)) -> (z = w -> A.x(x = y -> z = w)))
16	15	exp32 377	. . 3 |- (A.x(x = z /\ x = w) -> (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w)))))
17	1, 16	sylbir 199	. 2 |- ((A.x x = z /\ A.x x = w) -> (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w)))))
18		equequ1 1171	. . . . . . 7 |- (x = y -> (x = w <-> y = w))
19	18	ad2antll 407	. . . . . 6 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (x = w <-> y = w))
20		ax-12 1004	. . . . . . . . 9 |- (-. A.x x = y -> (-. A.x x = w -> (y = w -> A.x y = w)))
21	20	impcom 349	. . . . . . . 8 |- ((-. A.x x = w /\ -. A.x x = y) -> (y = w -> A.x y = w))
22	21	adantrr 395	. . . . . . 7 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (y = w -> A.x y = w))
23		equtrr 1169	. . . . . . . 8 |- (y = w -> (x = y -> x = w))
24	23	19.20i 1028	. . . . . . 7 |- (A.x y = w -> A.x(x = y -> x = w))
25	22, 24	syl6 22	. . . . . 6 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (y = w -> A.x(x = y -> x = w)))
26	19, 25	sylbid 201	. . . . 5 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (x = w -> A.x(x = y -> x = w)))
27	26	adantll 392	. . . 4 |- (((A.x x = z /\ -. A.x x = w) /\ (-. A.x x = y /\ x = y)) -> (x = w -> A.x(x = y -> x = w)))
28		equequ1 1171	. . . . . . 7 |- (x = z -> (x = w <-> z = w))
29	28	a4s 1020	. . . . . 6 |- (A.x x = z -> (x = w <-> z = w))
30	29	imbi2d 615	. . . . . . 7 |- (A.x x = z -> ((x = y -> x = w) <-> (x = y -> z = w)))
31	30	dral2 1192	. . . . . 6 |- (A.x x = z -> (A.x(x = y -> x = w) <-> A.x(x = y -> z = w)))
32	29, 31	imbi12d 629	. . . . 5 |- (A.x x = z -> ((x = w -> A.x(x = y -> x = w)) <-> (z = w -> A.x(x = y -> z = w))))
33	32	ad2antrr 404	. . . 4 |- (((A.x x = z /\ -. A.x x = w) /\ (-. A.x x = y /\ x = y)) -> ((x = w -> A.x(x = y -> x = w)) <-> (z = w -> A.x(x = y -> z = w))))
34	27, 33	mpbid 193	. . 3 |- (((A.x x = z /\ -. A.x x = w) /\ (-. A.x x = y /\ x = y)) -> (z = w -> A.x(x = y -> z = w)))
35	34	exp32 377	. 2 |- ((A.x x = z /\ -. A.x x = w) -> (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w)))))
36		equequ2 1172	. . . . . . 7 |- (x = y -> (z = x <-> z = y))
37	36	ad2antll 407	. . . . . 6 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = x <-> z = y))
38		ax-12 1004	. . . . . . . . 9 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
39	38	imp 348	. . . . . . . 8 |- ((-. A.x x = z /\ -. A.x x = y) -> (z = y -> A.x z = y))
40	39	adantrr 395	. . . . . . 7 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = y -> A.x z = y))
41	36	biimprcd 154	. . . . . . . 8 |- (z = y -> (x = y -> z = x))
42	41	19.20i 1028	. . . . . . 7 |- (A.x z = y -> A.x(x = y -> z = x))
43	40, 42	syl6 22	. . . . . 6 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = y -> A.x(x = y -> z = x)))
44	37, 43	sylbid 201	. . . . 5 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = x -> A.x(x = y -> z = x)))
45	44	adantlr 393	. . . 4 |- (((-. A.x x = z /\ A.x x = w) /\ (-. A.x x = y /\ x = y)) -> (z = x -> A.x(x = y -> z = x)))
46	7	a4s 1020	. . . . . 6 |- (A.x x = w -> (z = x <-> z = w))
47	46	imbi2d 615	. . . . . . 7 |- (A.x x = w -> ((x = y -> z = x) <-> (x = y -> z = w)))
48	47	dral2 1192	. . . . . 6 |- (A.x x = w -> (A.x(x = y -> z = x) <-> A.x(x = y -> z = w)))
49	46, 48	imbi12d 629	. . . . 5 |- (A.x x = w -> ((z = x -> A.x(x = y -> z = x)) <-> (z = w -> A.x(x = y -> z = w))))
50	49	ad2antlr 405	. . . 4 |- (((-. A.x x = z /\ A.x x = w) /\ (-. A.x x = y /\ x = y)) -> ((z = x -> A.x(x = y -> z = x)) <-> (z = w -> A.x(x = y -> z = w))))
51	45, 50	mpbid 193	. . 3 |- (((-. A.x x = z /\ A.x x = w) /\ (-. A.x x = y /\ x = y)) -> (z = w -> A.x(x = y -> z = w)))
52	51	exp32 377	. 2 |- ((-. A.x x = z /\ A.x x = w) -> (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w)))))
53		a9e 1161	. . . . 5 |- E.u u = w
54		a9e 1161	. . . . . . 7 |- E.v v = z
55		ax-1 4	. . . . . . . . . . 11 |- (v = u -> (x = y -> v = u))
56	55	19.21aiv 1324	. . . . . . . . . 10 |- (v = u -> A.x(x = y -> v = u))
57		equequ1 1171	. . . . . . . . . . . . 13 |- (v = z -> (v = u <-> z = u))
58		equequ2 1172	. . . . . . . . . . . . 13 |- (u = w -> (z = u <-> z = w))
59	57, 58	sylan9bb 543	. . . . . . . . . . . 12 |- ((v = z /\ u = w) -> (v = u <-> z = w))
60	59	adantl 388	. . . . . . . . . . 11 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> (v = u <-> z = w))
61		dveeq2 1249	. . . . . . . . . . . . . . 15 |- (-. A.x x = z -> (v = z -> A.x v = z))
62		dveeq2 1249	. . . . . . . . . . . . . . 15 |- (-. A.x x = w -> (u = w -> A.x u = w))
63	61, 62	im2anan9 566	. . . . . . . . . . . . . 14 |- ((-. A.x x = z /\ -. A.x x = w) -> ((v = z /\ u = w) -> (A.x v = z /\ A.x u = w)))
64	63	imp 348	. . . . . . . . . . . . 13 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> (A.x v = z /\ A.x u = w))
65		19.26 1103	. . . . . . . . . . . . 13 |- (A.x(v = z /\ u = w) <-> (A.x v = z /\ A.x u = w))
66	64, 65	sylibr 198	. . . . . . . . . . . 12 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> A.x(v = z /\ u = w))
67		hba1 1039	. . . . . . . . . . . . 13 |- (A.x(v = z /\ u = w) -> A.xA.x(v = z /\ u = w))
68	59	a4s 1020	. . . . . . . . . . . . . 14 |- (A.x(v = z /\ u = w) -> (v = u <-> z = w))
69	68	imbi2d 615	. . . . . . . . . . . . 13 |- (A.x(v = z /\ u = w) -> ((x = y -> v = u) <-> (x = y -> z = w)))
70	67, 69	albid 1140	. . . . . . . . . . . 12 |- (A.x(v = z /\ u = w) -> (A.x(x = y -> v = u) <-> A.x(x = y -> z = w)))
71	66, 70	syl 10	. . . . . . . . . . 11 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> (A.x(x = y -> v = u) <-> A.x(x = y -> z = w)))
72	60, 71	imbi12d 629	. . . . . . . . . 10 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> ((v = u -> A.x(x = y -> v = u)) <-> (z = w -> A.x(x = y -> z = w))))
73	56, 72	mpbii 191	. . . . . . . . 9 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> (z = w -> A.x(x = y -> z = w)))
74	73	exp32 377	. . . . . . . 8 |- ((-. A.x x = z /\ -. A.x x = w) -> (v = z -> (u = w -> (z = w -> A.x(x = y -> z = w)))))
75	74	19.23adv 1251	. . . . . . 7 |- ((-. A.x x = z /\ -. A.x x = w) -> (E.v v = z -> (u = w -> (z = w -> A.x(x = y -> z = w)))))
76	54, 75	mpi 44	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = w) -> (u = w -> (z = w -> A.x(x = y -> z = w))))
77	76	19.23adv 1251	. . . . 5 |- ((-. A.x x = z /\ -. A.x x = w) -> (E.u u = w -> (z = w -> A.x(x = y -> z = w))))
78	53, 77	mpi 44	. . . 4 |- ((-. A.x x = z /\ -. A.x x = w) -> (z = w -> A.x(x = y -> z = w)))
79	78	a1d 12	. . 3 |- ((-. A.x x = z /\ -. A.x x = w) -> (x = y -> (z = w -> A.x(x = y -> z = w))))
80	79	a1d 12	. 2 |- ((-. A.x x = z /\ -. A.x x = w) -> (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w)))))
81	17, 35, 52, 80	4cases 763	1 |- (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017

Copyright terms: Public domain