Theorem ax11indalem 2197

Description: Lemma for ax11inda2 2199 and ax11inda 2200. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax11indalem.1	⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Assertion

Ref	Expression
ax11indalem	⊢ (¬ ∀y y = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))

Proof of Theorem ax11indalem

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . . . . 9 ⊢ (∀xφ → (x = y → ∀xφ))
2	1	a5i-o 2150	. . . . . . . 8 ⊢ (∀xφ → ∀x(x = y → ∀xφ))
3	2	a1i 10	. . . . . . 7 ⊢ (∀z z = x → (∀xφ → ∀x(x = y → ∀xφ)))
4		biidd 228	. . . . . . . 8 ⊢ (∀z z = x → (φ ↔ φ))
5	4	dral1-o 2154	. . . . . . 7 ⊢ (∀z z = x → (∀zφ ↔ ∀xφ))
6	5	imbi2d 307	. . . . . . . 8 ⊢ (∀z z = x → ((x = y → ∀zφ) ↔ (x = y → ∀xφ)))
7	6	dral2-o 2181	. . . . . . 7 ⊢ (∀z z = x → (∀x(x = y → ∀zφ) ↔ ∀x(x = y → ∀xφ)))
8	3, 5, 7	3imtr4d 259	. . . . . 6 ⊢ (∀z z = x → (∀zφ → ∀x(x = y → ∀zφ)))
9	8	aecoms-o 2152	. . . . 5 ⊢ (∀x x = z → (∀zφ → ∀x(x = y → ∀zφ)))
10	9	a1d 22	. . . 4 ⊢ (∀x x = z → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))
11	10	a1d 22	. . 3 ⊢ (∀x x = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
12	11	adantr 451	. 2 ⊢ ((∀x x = z ∧ ¬ ∀y y = z) → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
13		simplr 731	. . . . 5 ⊢ ((((¬ ∀x x = z ∧ ¬ ∀y y = z) ∧ ¬ ∀x x = y) ∧ x = y) → ¬ ∀x x = y)
14		aecom-o 2151	. . . . . . . . 9 ⊢ (∀z z = x → ∀x x = z)
15	14	con3i 127	. . . . . . . 8 ⊢ (¬ ∀x x = z → ¬ ∀z z = x)
16		aecom-o 2151	. . . . . . . . 9 ⊢ (∀z z = y → ∀y y = z)
17	16	con3i 127	. . . . . . . 8 ⊢ (¬ ∀y y = z → ¬ ∀z z = y)
18		ax12o 1934	. . . . . . . . 9 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
19	18	imp 418	. . . . . . . 8 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → (x = y → ∀z x = y))
20	15, 17, 19	syl2an 463	. . . . . . 7 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → (x = y → ∀z x = y))
21	20	imp 418	. . . . . 6 ⊢ (((¬ ∀x x = z ∧ ¬ ∀y y = z) ∧ x = y) → ∀z x = y)
22	21	adantlr 695	. . . . 5 ⊢ ((((¬ ∀x x = z ∧ ¬ ∀y y = z) ∧ ¬ ∀x x = y) ∧ x = y) → ∀z x = y)
23		hbnae-o 2179	. . . . . . 7 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
24		hba1-o 2149	. . . . . . 7 ⊢ (∀z x = y → ∀z∀z x = y)
25	23, 24	hban 1828	. . . . . 6 ⊢ ((¬ ∀x x = y ∧ ∀z x = y) → ∀z(¬ ∀x x = y ∧ ∀z x = y))
26		ax-4 2135	. . . . . . 7 ⊢ (∀z x = y → x = y)
27		ax11indalem.1	. . . . . . . 8 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
28	27	imp 418	. . . . . . 7 ⊢ ((¬ ∀x x = y ∧ x = y) → (φ → ∀x(x = y → φ)))
29	26, 28	sylan2 460	. . . . . 6 ⊢ ((¬ ∀x x = y ∧ ∀z x = y) → (φ → ∀x(x = y → φ)))
30	25, 29	alimdh 1563	. . . . 5 ⊢ ((¬ ∀x x = y ∧ ∀z x = y) → (∀zφ → ∀z∀x(x = y → φ)))
31	13, 22, 30	syl2anc 642	. . . 4 ⊢ ((((¬ ∀x x = z ∧ ¬ ∀y y = z) ∧ ¬ ∀x x = y) ∧ x = y) → (∀zφ → ∀z∀x(x = y → φ)))
32		ax-7 1734	. . . . . 6 ⊢ (∀z∀x(x = y → φ) → ∀x∀z(x = y → φ))
33		hbnae-o 2179	. . . . . . . 8 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
34		hbnae-o 2179	. . . . . . . 8 ⊢ (¬ ∀y y = z → ∀x ¬ ∀y y = z)
35	33, 34	hban 1828	. . . . . . 7 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → ∀x(¬ ∀x x = z ∧ ¬ ∀y y = z))
36		hbnae-o 2179	. . . . . . . . . 10 ⊢ (¬ ∀x x = z → ∀z ¬ ∀x x = z)
37		hbnae-o 2179	. . . . . . . . . 10 ⊢ (¬ ∀y y = z → ∀z ¬ ∀y y = z)
38	36, 37	hban 1828	. . . . . . . . 9 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → ∀z(¬ ∀x x = z ∧ ¬ ∀y y = z))
39	38, 20	nfdh 1767	. . . . . . . 8 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → Ⅎz x = y)
40		19.21t 1795	. . . . . . . 8 ⊢ (Ⅎz x = y → (∀z(x = y → φ) ↔ (x = y → ∀zφ)))
41	39, 40	syl 15	. . . . . . 7 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → (∀z(x = y → φ) ↔ (x = y → ∀zφ)))
42	35, 41	albidh 1590	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → (∀x∀z(x = y → φ) ↔ ∀x(x = y → ∀zφ)))
43	32, 42	syl5ib 210	. . . . 5 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → (∀z∀x(x = y → φ) → ∀x(x = y → ∀zφ)))
44	43	ad2antrr 706	. . . 4 ⊢ ((((¬ ∀x x = z ∧ ¬ ∀y y = z) ∧ ¬ ∀x x = y) ∧ x = y) → (∀z∀x(x = y → φ) → ∀x(x = y → ∀zφ)))
45	31, 44	syld 40	. . 3 ⊢ ((((¬ ∀x x = z ∧ ¬ ∀y y = z) ∧ ¬ ∀x x = y) ∧ x = y) → (∀zφ → ∀x(x = y → ∀zφ)))
46	45	exp31 587	. 2 ⊢ ((¬ ∀x x = z ∧ ¬ ∀y y = z) → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
47	12, 46	pm2.61ian 765	1 ⊢ (¬ ∀y y = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544

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:  ax11inda2  2199