Theorem ax11inda2ALT 2198

Description: A proof of ax11inda2 2199 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax11inda2.1	⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Assertion

Ref	Expression
ax11inda2ALT	⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))

Distinct variable group:   y,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem ax11inda2ALT

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . . . 8 ⊢ (∀xφ → (x = y → ∀xφ))
2	1	a5i-o 2150	. . . . . . 7 ⊢ (∀xφ → ∀x(x = y → ∀xφ))
3	2	a1i 10	. . . . . 6 ⊢ (∀z z = x → (∀xφ → ∀x(x = y → ∀xφ)))
4		biidd 228	. . . . . . 7 ⊢ (∀z z = x → (φ ↔ φ))
5	4	dral1-o 2154	. . . . . 6 ⊢ (∀z z = x → (∀zφ ↔ ∀xφ))
6	5	imbi2d 307	. . . . . . 7 ⊢ (∀z z = x → ((x = y → ∀zφ) ↔ (x = y → ∀xφ)))
7	6	dral2-o 2181	. . . . . 6 ⊢ (∀z z = x → (∀x(x = y → ∀zφ) ↔ ∀x(x = y → ∀xφ)))
8	3, 5, 7	3imtr4d 259	. . . . 5 ⊢ (∀z z = x → (∀zφ → ∀x(x = y → ∀zφ)))
9	8	aecoms-o 2152	. . . 4 ⊢ (∀x x = z → (∀zφ → ∀x(x = y → ∀zφ)))
10	9	a1d 22	. . 3 ⊢ (∀x x = z → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))
11	10	a1d 22	. 2 ⊢ (∀x x = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
12		simplr 731	. . . . 5 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = y) ∧ x = y) → ¬ ∀x x = y)
13		dveeq1-o 2187	. . . . . . . 8 ⊢ (¬ ∀z z = x → (x = y → ∀z x = y))
14	13	naecoms-o 2178	. . . . . . 7 ⊢ (¬ ∀x x = z → (x = y → ∀z x = y))
15	14	imp 418	. . . . . 6 ⊢ ((¬ ∀x x = z ∧ x = y) → ∀z x = y)
16	15	adantlr 695	. . . . 5 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = y) ∧ x = y) → ∀z x = y)
17		hbnae-o 2179	. . . . . . 7 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
18		hba1-o 2149	. . . . . . 7 ⊢ (∀z x = y → ∀z∀z x = y)
19	17, 18	hban 1828	. . . . . 6 ⊢ ((¬ ∀x x = y ∧ ∀z x = y) → ∀z(¬ ∀x x = y ∧ ∀z x = y))
20		ax-4 2135	. . . . . . 7 ⊢ (∀z x = y → x = y)
21		ax11inda2.1	. . . . . . . 8 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
22	21	imp 418	. . . . . . 7 ⊢ ((¬ ∀x x = y ∧ x = y) → (φ → ∀x(x = y → φ)))
23	20, 22	sylan2 460	. . . . . 6 ⊢ ((¬ ∀x x = y ∧ ∀z x = y) → (φ → ∀x(x = y → φ)))
24	19, 23	alimdh 1563	. . . . 5 ⊢ ((¬ ∀x x = y ∧ ∀z x = y) → (∀zφ → ∀z∀x(x = y → φ)))
25	12, 16, 24	syl2anc 642	. . . 4 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = y) ∧ x = y) → (∀zφ → ∀z∀x(x = y → φ)))
26		ax-7 1734	. . . . . 6 ⊢ (∀z∀x(x = y → φ) → ∀x∀z(x = y → φ))
27		hbnae-o 2179	. . . . . . 7 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
28		hbnae-o 2179	. . . . . . . . 9 ⊢ (¬ ∀x x = z → ∀z ¬ ∀x x = z)
29	28, 14	nfdh 1767	. . . . . . . 8 ⊢ (¬ ∀x x = z → Ⅎz x = y)
30		19.21t 1795	. . . . . . . 8 ⊢ (Ⅎz x = y → (∀z(x = y → φ) ↔ (x = y → ∀zφ)))
31	29, 30	syl 15	. . . . . . 7 ⊢ (¬ ∀x x = z → (∀z(x = y → φ) ↔ (x = y → ∀zφ)))
32	27, 31	albidh 1590	. . . . . 6 ⊢ (¬ ∀x x = z → (∀x∀z(x = y → φ) ↔ ∀x(x = y → ∀zφ)))
33	26, 32	syl5ib 210	. . . . 5 ⊢ (¬ ∀x x = z → (∀z∀x(x = y → φ) → ∀x(x = y → ∀zφ)))
34	33	ad2antrr 706	. . . 4 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = y) ∧ x = y) → (∀z∀x(x = y → φ) → ∀x(x = y → ∀zφ)))
35	25, 34	syld 40	. . 3 ⊢ (((¬ ∀x x = z ∧ ¬ ∀x x = y) ∧ x = y) → (∀zφ → ∀x(x = y → ∀zφ)))
36	35	exp31 587	. 2 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
37	11, 36	pm2.61i 156	1 ⊢ (¬ ∀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: (None)