Theorem ax11eq 2193

Description: Basis step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Atomic formula for equality predicate. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ax11eq

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.26 1593	. . 3 |- (A.x(x = z /\ x = w) <-> (A.x x = z /\ A.x x = w))
2		equid 1676	. . . . . . . 8 |- x = x
3	2	a1i 10	. . . . . . 7 |- (x = y -> x = x)
4	3	ax-gen 1546	. . . . . 6 |- A.x(x = y -> x = x)
5	4	a1i 10	. . . . 5 |- (x = x -> A.x(x = y -> x = x))
6		equequ1 1684	. . . . . . . . 9 |- (x = z -> (x = x <-> z = x))
7		equequ2 1686	. . . . . . . . 9 |- (x = w -> (z = x <-> z = w))
8	6, 7	sylan9bb 680	. . . . . . . 8 |- ((x = z /\ x = w) -> (x = x <-> z = w))
9	8	sps-o 2159	. . . . . . 7 |- (A.x(x = z /\ x = w) -> (x = x <-> z = w))
10		nfa1-o 2166	. . . . . . . 8 |- F/xA.x(x = z /\ x = w)
11	9	imbi2d 307	. . . . . . . 8 |- (A.x(x = z /\ x = w) -> ((x = y -> x = x) <-> (x = y -> z = w)))
12	10, 11	albid 1772	. . . . . . 7 |- (A.x(x = z /\ x = w) -> (A.x(x = y -> x = x) <-> A.x(x = y -> z = w)))
13	9, 12	imbi12d 311	. . . . . 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 451	. . . . 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 202	. . . 4 |- ((A.x(x = z /\ x = w) /\ (-. A.x x = y /\ x = y)) -> (z = w -> A.x(x = y -> z = w)))
16	15	exp32 588	. . 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 204	. 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 1684	. . . . . . 7 |- (x = y -> (x = w <-> y = w))
19	18	ad2antll 709	. . . . . 6 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (x = w <-> y = w))
20		ax12o 1934	. . . . . . . . 9 |- (-. A.x x = y -> (-. A.x x = w -> (y = w -> A.x y = w)))
21	20	impcom 419	. . . . . . . 8 |- ((-. A.x x = w /\ -. A.x x = y) -> (y = w -> A.x y = w))
22	21	adantrr 697	. . . . . . 7 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (y = w -> A.x y = w))
23		equtrr 1683	. . . . . . . 8 |- (y = w -> (x = y -> x = w))
24	23	alimi 1559	. . . . . . 7 |- (A.x y = w -> A.x(x = y -> x = w))
25	22, 24	syl6 29	. . . . . 6 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (y = w -> A.x(x = y -> x = w)))
26	19, 25	sylbid 206	. . . . 5 |- ((-. A.x x = w /\ (-. A.x x = y /\ x = y)) -> (x = w -> A.x(x = y -> x = w)))
27	26	adantll 694	. . . 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 1684	. . . . . . 7 |- (x = z -> (x = w <-> z = w))
29	28	sps-o 2159	. . . . . 6 |- (A.x x = z -> (x = w <-> z = w))
30	29	imbi2d 307	. . . . . . 7 |- (A.x x = z -> ((x = y -> x = w) <-> (x = y -> z = w)))
31	30	dral2-o 2181	. . . . . 6 |- (A.x x = z -> (A.x(x = y -> x = w) <-> A.x(x = y -> z = w)))
32	29, 31	imbi12d 311	. . . . 5 |- (A.x x = z -> ((x = w -> A.x(x = y -> x = w)) <-> (z = w -> A.x(x = y -> z = w))))
33	32	ad2antrr 706	. . . 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 201	. . 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 588	. 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 1686	. . . . . . 7 |- (x = y -> (z = x <-> z = y))
37	36	ad2antll 709	. . . . . 6 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = x <-> z = y))
38		ax12o 1934	. . . . . . . . 9 |- (-. A.x x = z -> (-. A.x x = y -> (z = y -> A.x z = y)))
39	38	imp 418	. . . . . . . 8 |- ((-. A.x x = z /\ -. A.x x = y) -> (z = y -> A.x z = y))
40	39	adantrr 697	. . . . . . 7 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = y -> A.x z = y))
41	36	biimprcd 216	. . . . . . . 8 |- (z = y -> (x = y -> z = x))
42	41	alimi 1559	. . . . . . 7 |- (A.x z = y -> A.x(x = y -> z = x))
43	40, 42	syl6 29	. . . . . 6 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = y -> A.x(x = y -> z = x)))
44	37, 43	sylbid 206	. . . . 5 |- ((-. A.x x = z /\ (-. A.x x = y /\ x = y)) -> (z = x -> A.x(x = y -> z = x)))
45	44	adantlr 695	. . . 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	sps-o 2159	. . . . . 6 |- (A.x x = w -> (z = x <-> z = w))
47	46	imbi2d 307	. . . . . . 7 |- (A.x x = w -> ((x = y -> z = x) <-> (x = y -> z = w)))
48	47	dral2-o 2181	. . . . . 6 |- (A.x x = w -> (A.x(x = y -> z = x) <-> A.x(x = y -> z = w)))
49	46, 48	imbi12d 311	. . . . 5 |- (A.x x = w -> ((z = x -> A.x(x = y -> z = x)) <-> (z = w -> A.x(x = y -> z = w))))
50	49	ad2antlr 707	. . . 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 201	. . 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 588	. 2 |- ((-. A.x x = z /\ A.x x = w) -> (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w)))))
53		a9ev 1656	. . . . 5 |- E.u u = w
54		a9ev 1656	. . . . . . 7 |- E.v v = z
55		ax-1 6	. . . . . . . . . . 11 |- (v = u -> (x = y -> v = u))
56	55	alrimiv 1631	. . . . . . . . . 10 |- (v = u -> A.x(x = y -> v = u))
57		equequ1 1684	. . . . . . . . . . . . 13 |- (v = z -> (v = u <-> z = u))
58		equequ2 1686	. . . . . . . . . . . . 13 |- (u = w -> (z = u <-> z = w))
59	57, 58	sylan9bb 680	. . . . . . . . . . . 12 |- ((v = z /\ u = w) -> (v = u <-> z = w))
60	59	adantl 452	. . . . . . . . . . 11 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> (v = u <-> z = w))
61		dveeq2-o 2184	. . . . . . . . . . . . . . 15 |- (-. A.x x = z -> (v = z -> A.x v = z))
62		dveeq2-o 2184	. . . . . . . . . . . . . . 15 |- (-. A.x x = w -> (u = w -> A.x u = w))
63	61, 62	im2anan9 808	. . . . . . . . . . . . . 14 |- ((-. A.x x = z /\ -. A.x x = w) -> ((v = z /\ u = w) -> (A.x v = z /\ A.x u = w)))
64	63	imp 418	. . . . . . . . . . . . 13 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> (A.x v = z /\ A.x u = w))
65		19.26 1593	. . . . . . . . . . . . 13 |- (A.x(v = z /\ u = w) <-> (A.x v = z /\ A.x u = w))
66	64, 65	sylibr 203	. . . . . . . . . . . 12 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> A.x(v = z /\ u = w))
67		nfa1-o 2166	. . . . . . . . . . . . 13 |- F/xA.x(v = z /\ u = w)
68	59	sps-o 2159	. . . . . . . . . . . . . 14 |- (A.x(v = z /\ u = w) -> (v = u <-> z = w))
69	68	imbi2d 307	. . . . . . . . . . . . 13 |- (A.x(v = z /\ u = w) -> ((x = y -> v = u) <-> (x = y -> z = w)))
70	67, 69	albid 1772	. . . . . . . . . . . 12 |- (A.x(v = z /\ u = w) -> (A.x(x = y -> v = u) <-> A.x(x = y -> z = w)))
71	66, 70	syl 15	. . . . . . . . . . 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 311	. . . . . . . . . 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 202	. . . . . . . . 9 |- (((-. A.x x = z /\ -. A.x x = w) /\ (v = z /\ u = w)) -> (z = w -> A.x(x = y -> z = w)))
74	73	exp32 588	. . . . . . . 8 |- ((-. A.x x = z /\ -. A.x x = w) -> (v = z -> (u = w -> (z = w -> A.x(x = y -> z = w)))))
75	74	exlimdv 1636	. . . . . . 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 16	. . . . . 6 |- ((-. A.x x = z /\ -. A.x x = w) -> (u = w -> (z = w -> A.x(x = y -> z = w))))
77	76	exlimdv 1636	. . . . 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 16	. . . 4 |- ((-. A.x x = z /\ -. A.x x = w) -> (z = w -> A.x(x = y -> z = w)))
79	78	a1d 22	. . 3 |- ((-. A.x x = z /\ -. A.x x = w) -> (x = y -> (z = w -> A.x(x = y -> z = w))))
80	79	a1d 22	. 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 915	1 |- (-. A.x x = y -> (x = y -> (z = w -> A.x(x = y -> z = w))))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)