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	⊢ (¬ ∀x x = y → (x = y → (z = w → ∀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 ⊢ (∀x(x = z ∧ x = w) ↔ (∀x x = z ∧ ∀x x = w))
2		equid 1676	. . . . . . . 8 ⊢ x = x
3	2	a1i 10	. . . . . . 7 ⊢ (x = y → x = x)
4	3	ax-gen 1546	. . . . . 6 ⊢ ∀x(x = y → x = x)
5	4	a1i 10	. . . . 5 ⊢ (x = x → ∀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 ⊢ (∀x(x = z ∧ x = w) → (x = x ↔ z = w))
10		nfa1-o 2166	. . . . . . . 8 ⊢ Ⅎx∀x(x = z ∧ x = w)
11	9	imbi2d 307	. . . . . . . 8 ⊢ (∀x(x = z ∧ x = w) → ((x = y → x = x) ↔ (x = y → z = w)))
12	10, 11	albid 1772	. . . . . . 7 ⊢ (∀x(x = z ∧ x = w) → (∀x(x = y → x = x) ↔ ∀x(x = y → z = w)))
13	9, 12	imbi12d 311	. . . . . 6 ⊢ (∀x(x = z ∧ x = w) → ((x = x → ∀x(x = y → x = x)) ↔ (z = w → ∀x(x = y → z = w))))
14	13	adantr 451	. . . . 5 ⊢ ((∀x(x = z ∧ x = w) ∧ (¬ ∀x x = y ∧ x = y)) → ((x = x → ∀x(x = y → x = x)) ↔ (z = w → ∀x(x = y → z = w))))
15	5, 14	mpbii 202	. . . 4 ⊢ ((∀x(x = z ∧ x = w) ∧ (¬ ∀x x = y ∧ x = y)) → (z = w → ∀x(x = y → z = w)))
16	15	exp32 588	. . 3 ⊢ (∀x(x = z ∧ x = w) → (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w)))))
17	1, 16	sylbir 204	. 2 ⊢ ((∀x x = z ∧ ∀x x = w) → (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w)))))
18		equequ1 1684	. . . . . . 7 ⊢ (x = y → (x = w ↔ y = w))
19	18	ad2antll 709	. . . . . 6 ⊢ ((¬ ∀x x = w ∧ (¬ ∀x x = y ∧ x = y)) → (x = w ↔ y = w))
20		ax12o 1934	. . . . . . . . 9 ⊢ (¬ ∀x x = y → (¬ ∀x x = w → (y = w → ∀x y = w)))
21	20	impcom 419	. . . . . . . 8 ⊢ ((¬ ∀x x = w ∧ ¬ ∀x x = y) → (y = w → ∀x y = w))
22	21	adantrr 697	. . . . . . 7 ⊢ ((¬ ∀x x = w ∧ (¬ ∀x x = y ∧ x = y)) → (y = w → ∀x y = w))
23		equtrr 1683	. . . . . . . 8 ⊢ (y = w → (x = y → x = w))
24	23	alimi 1559	. . . . . . 7 ⊢ (∀x y = w → ∀x(x = y → x = w))
25	22, 24	syl6 29	. . . . . 6 ⊢ ((¬ ∀x x = w ∧ (¬ ∀x x = y ∧ x = y)) → (y = w → ∀x(x = y → x = w)))
26	19, 25	sylbid 206	. . . . 5 ⊢ ((¬ ∀x x = w ∧ (¬ ∀x x = y ∧ x = y)) → (x = w → ∀x(x = y → x = w)))
27	26	adantll 694	. . . 4 ⊢ (((∀x x = z ∧ ¬ ∀x x = w) ∧ (¬ ∀x x = y ∧ x = y)) → (x = w → ∀x(x = y → x = w)))
28		equequ1 1684	. . . . . . 7 ⊢ (x = z → (x = w ↔ z = w))
29	28	sps-o 2159	. . . . . 6 ⊢ (∀x x = z → (x = w ↔ z = w))
30	29	imbi2d 307	. . . . . . 7 ⊢ (∀x x = z → ((x = y → x = w) ↔ (x = y → z = w)))
31	30	dral2-o 2181	. . . . . 6 ⊢ (∀x x = z → (∀x(x = y → x = w) ↔ ∀x(x = y → z = w)))
32	29, 31	imbi12d 311	. . . . 5 ⊢ (∀x x = z → ((x = w → ∀x(x = y → x = w)) ↔ (z = w → ∀x(x = y → z = w))))
33	32	ad2antrr 706	. . . 4 ⊢ (((∀x x = z ∧ ¬ ∀x x = w) ∧ (¬ ∀x x = y ∧ x = y)) → ((x = w → ∀x(x = y → x = w)) ↔ (z = w → ∀x(x = y → z = w))))
34	27, 33	mpbid 201	. . 3 ⊢ (((∀x x = z ∧ ¬ ∀x x = w) ∧ (¬ ∀x x = y ∧ x = y)) → (z = w → ∀x(x = y → z = w)))
35	34	exp32 588	. 2 ⊢ ((∀x x = z ∧ ¬ ∀x x = w) → (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w)))))
36		equequ2 1686	. . . . . . 7 ⊢ (x = y → (z = x ↔ z = y))
37	36	ad2antll 709	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ (¬ ∀x x = y ∧ x = y)) → (z = x ↔ z = y))
38		ax12o 1934	. . . . . . . . 9 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (z = y → ∀x z = y)))
39	38	imp 418	. . . . . . . 8 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = y) → (z = y → ∀x z = y))
40	39	adantrr 697	. . . . . . 7 ⊢ ((¬ ∀x x = z ∧ (¬ ∀x x = y ∧ x = y)) → (z = y → ∀x z = y))
41	36	biimprcd 216	. . . . . . . 8 ⊢ (z = y → (x = y → z = x))
42	41	alimi 1559	. . . . . . 7 ⊢ (∀x z = y → ∀x(x = y → z = x))
43	40, 42	syl6 29	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ (¬ ∀x x = y ∧ x = y)) → (z = y → ∀x(x = y → z = x)))
44	37, 43	sylbid 206	. . . . 5 ⊢ ((¬ ∀x x = z ∧ (¬ ∀x x = y ∧ x = y)) → (z = x → ∀x(x = y → z = x)))
45	44	adantlr 695	. . . 4 ⊢ (((¬ ∀x x = z ∧ ∀x x = w) ∧ (¬ ∀x x = y ∧ x = y)) → (z = x → ∀x(x = y → z = x)))
46	7	sps-o 2159	. . . . . 6 ⊢ (∀x x = w → (z = x ↔ z = w))
47	46	imbi2d 307	. . . . . . 7 ⊢ (∀x x = w → ((x = y → z = x) ↔ (x = y → z = w)))
48	47	dral2-o 2181	. . . . . 6 ⊢ (∀x x = w → (∀x(x = y → z = x) ↔ ∀x(x = y → z = w)))
49	46, 48	imbi12d 311	. . . . 5 ⊢ (∀x x = w → ((z = x → ∀x(x = y → z = x)) ↔ (z = w → ∀x(x = y → z = w))))
50	49	ad2antlr 707	. . . 4 ⊢ (((¬ ∀x x = z ∧ ∀x x = w) ∧ (¬ ∀x x = y ∧ x = y)) → ((z = x → ∀x(x = y → z = x)) ↔ (z = w → ∀x(x = y → z = w))))
51	45, 50	mpbid 201	. . 3 ⊢ (((¬ ∀x x = z ∧ ∀x x = w) ∧ (¬ ∀x x = y ∧ x = y)) → (z = w → ∀x(x = y → z = w)))
52	51	exp32 588	. 2 ⊢ ((¬ ∀x x = z ∧ ∀x x = w) → (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w)))))
53		a9ev 1656	. . . . 5 ⊢ ∃u u = w
54		a9ev 1656	. . . . . . 7 ⊢ ∃v v = z
55		ax-1 6	. . . . . . . . . . 11 ⊢ (v = u → (x = y → v = u))
56	55	alrimiv 1631	. . . . . . . . . 10 ⊢ (v = u → ∀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 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = w) ∧ (v = z ∧ u = w)) → (v = u ↔ z = w))
61		dveeq2-o 2184	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀x x = z → (v = z → ∀x v = z))
62		dveeq2-o 2184	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀x x = w → (u = w → ∀x u = w))
63	61, 62	im2anan9 808	. . . . . . . . . . . . . 14 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → ((v = z ∧ u = w) → (∀x v = z ∧ ∀x u = w)))
64	63	imp 418	. . . . . . . . . . . . 13 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = w) ∧ (v = z ∧ u = w)) → (∀x v = z ∧ ∀x u = w))
65		19.26 1593	. . . . . . . . . . . . 13 ⊢ (∀x(v = z ∧ u = w) ↔ (∀x v = z ∧ ∀x u = w))
66	64, 65	sylibr 203	. . . . . . . . . . . 12 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = w) ∧ (v = z ∧ u = w)) → ∀x(v = z ∧ u = w))
67		nfa1-o 2166	. . . . . . . . . . . . 13 ⊢ Ⅎx∀x(v = z ∧ u = w)
68	59	sps-o 2159	. . . . . . . . . . . . . 14 ⊢ (∀x(v = z ∧ u = w) → (v = u ↔ z = w))
69	68	imbi2d 307	. . . . . . . . . . . . 13 ⊢ (∀x(v = z ∧ u = w) → ((x = y → v = u) ↔ (x = y → z = w)))
70	67, 69	albid 1772	. . . . . . . . . . . 12 ⊢ (∀x(v = z ∧ u = w) → (∀x(x = y → v = u) ↔ ∀x(x = y → z = w)))
71	66, 70	syl 15	. . . . . . . . . . 11 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = w) ∧ (v = z ∧ u = w)) → (∀x(x = y → v = u) ↔ ∀x(x = y → z = w)))
72	60, 71	imbi12d 311	. . . . . . . . . 10 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = w) ∧ (v = z ∧ u = w)) → ((v = u → ∀x(x = y → v = u)) ↔ (z = w → ∀x(x = y → z = w))))
73	56, 72	mpbii 202	. . . . . . . . 9 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = w) ∧ (v = z ∧ u = w)) → (z = w → ∀x(x = y → z = w)))
74	73	exp32 588	. . . . . . . 8 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → (v = z → (u = w → (z = w → ∀x(x = y → z = w)))))
75	74	exlimdv 1636	. . . . . . 7 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → (∃v v = z → (u = w → (z = w → ∀x(x = y → z = w)))))
76	54, 75	mpi 16	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → (u = w → (z = w → ∀x(x = y → z = w))))
77	76	exlimdv 1636	. . . . 5 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → (∃u u = w → (z = w → ∀x(x = y → z = w))))
78	53, 77	mpi 16	. . . 4 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → (z = w → ∀x(x = y → z = w)))
79	78	a1d 22	. . 3 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → (x = y → (z = w → ∀x(x = y → z = w))))
80	79	a1d 22	. 2 ⊢ ((¬ ∀x x = z ∧ ¬ ∀x x = w) → (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w)))))
81	17, 35, 52, 80	4cases 915	1 ⊢ (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃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)