Theorem ax11inda2ALT 1368

Description: A proof of ax11inda2 1369 that is slightly more direct.

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

Proof of Theorem ax11inda2ALT

Step	Hyp	Ref	Expression
1		ax-1 4	. . . . . . . 8 ⊢ (∀xφ → (x = y → ∀xφ))
2	1	a5i 988	. . . . . . 7 ⊢ (∀xφ → ∀x(x = y → ∀xφ))
3	2	a1i 8	. . . . . 6 ⊢ (∀z z = x → (∀xφ → ∀x(x = y → ∀xφ)))
4		pm4.2i 171	. . . . . . 7 ⊢ (∀z z = x → (φ ↔ φ))
5	4	dral1 1153	. . . . . 6 ⊢ (∀z z = x → (∀zφ ↔ ∀xφ))
6	5	imbi2d 611	. . . . . . 7 ⊢ (∀z z = x → ((x = y → ∀zφ) ↔ (x = y → ∀xφ)))
7	6	dral2 1154	. . . . . 6 ⊢ (∀z z = x → (∀x(x = y → ∀zφ) ↔ ∀x(x = y → ∀xφ)))
8	3, 5, 7	3imtr4d 542	. . . . 5 ⊢ (∀z z = x → (∀zφ → ∀x(x = y → ∀zφ)))
9	8	alequcoms 1142	. . . 4 ⊢ (∀x x = z → (∀zφ → ∀x(x = y → ∀zφ)))
10	9	a1d 12	. . 3 ⊢ (∀x x = z → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))
11	10	a1d 12	. 2 ⊢ (∀x x = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
12		hbnae 1146	. . . . . . 7 ⊢ (¬ ∀x x = y → ∀z ¬ ∀x x = y)
13		hba1 1002	. . . . . . 7 ⊢ (∀z x = y → ∀z∀z x = y)
14	12, 13	hban 1008	. . . . . 6 ⊢ ((¬ ∀x x = y ⋀ ∀z x = y) → ∀z(¬ ∀x x = y ⋀ ∀z x = y))
15		ax11inda2.1	. . . . . . . 8 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
16	15	imp 350	. . . . . . 7 ⊢ ((¬ ∀x x = y ⋀ x = y) → (φ → ∀x(x = y → φ)))
17		ax-4 972	. . . . . . 7 ⊢ (∀z x = y → x = y)
18	16, 17	sylan2 451	. . . . . 6 ⊢ ((¬ ∀x x = y ⋀ ∀z x = y) → (φ → ∀x(x = y → φ)))
19	14, 18	19.20d 995	. . . . 5 ⊢ ((¬ ∀x x = y ⋀ ∀z x = y) → (∀zφ → ∀z∀x(x = y → φ)))
20		simplr 413	. . . . 5 ⊢ (((¬ ∀x x = z ⋀ ¬ ∀x x = y) ⋀ x = y) → ¬ ∀x x = y)
21		dveeq1 1353	. . . . . . . 8 ⊢ (¬ ∀z z = x → (x = y → ∀z x = y))
22	21	nalequcoms 1143	. . . . . . 7 ⊢ (¬ ∀x x = z → (x = y → ∀z x = y))
23	22	imp 350	. . . . . 6 ⊢ ((¬ ∀x x = z ⋀ x = y) → ∀z x = y)
24	23	adantlr 393	. . . . 5 ⊢ (((¬ ∀x x = z ⋀ ¬ ∀x x = y) ⋀ x = y) → ∀z x = y)
25	19, 20, 24	sylanc 471	. . . 4 ⊢ (((¬ ∀x x = z ⋀ ¬ ∀x x = y) ⋀ x = y) → (∀zφ → ∀z∀x(x = y → φ)))
26		hbnae 1146	. . . . . . 7 ⊢ (¬ ∀x x = z → ∀x ¬ ∀x x = z)
27		hbnae 1146	. . . . . . . . 9 ⊢ (¬ ∀x x = z → ∀z ¬ ∀x x = z)
28	27, 22	19.21ai 997	. . . . . . . 8 ⊢ (¬ ∀x x = z → ∀z(x = y → ∀z x = y))
29		19.21t 1114	. . . . . . . 8 ⊢ (∀z(x = y → ∀z x = y) → (∀z(x = y → φ) ↔ (x = y → ∀zφ)))
30	28, 29	syl 10	. . . . . . 7 ⊢ (¬ ∀x x = z → (∀z(x = y → φ) ↔ (x = y → ∀zφ)))
31	26, 30	albid 1103	. . . . . 6 ⊢ (¬ ∀x x = z → (∀x∀z(x = y → φ) ↔ ∀x(x = y → ∀zφ)))
32		ax-7 961	. . . . . 6 ⊢ (∀z∀x(x = y → φ) → ∀x∀z(x = y → φ))
33	31, 32	syl5bi 208	. . . . 5 ⊢ (¬ ∀x x = z → (∀z∀x(x = y → φ) → ∀x(x = y → ∀zφ)))
34	33	ad2antrr 404	. . . 4 ⊢ (((¬ ∀x x = z ⋀ ¬ ∀x x = y) ⋀ x = y) → (∀z∀x(x = y → φ) → ∀x(x = y → ∀zφ)))
35	25, 34	syld 27	. . 3 ⊢ (((¬ ∀x x = z ⋀ ¬ ∀x x = y) ⋀ x = y) → (∀zφ → ∀x(x = y → ∀zφ)))
36	35	exp31 376	. 2 ⊢ (¬ ∀x x = z → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
37	11, 36	pm2.61i 126	1 ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   = wceq 955

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139

This theorem depends on definitions:  df-bi 147  df-an 225

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