Theorem a12lem1 1374

Description: Proof of first hypothesis of a12study 1376.

Assertion

Ref	Expression
a12lem1	⊢ (¬ ∀z z = y → (∀z(z = x → z = y) → x = y))

Proof of Theorem a12lem1

Step	Hyp	Ref	Expression
1		equequ1 1132	. . . . . . 7 ⊢ (z = x → (z = x ↔ x = x))
2		equequ1 1132	. . . . . . 7 ⊢ (z = x → (z = y ↔ x = y))
3	1, 2	imbi12d 625	. . . . . 6 ⊢ (z = x → ((z = x → z = y) ↔ (x = x → x = y)))
4	3	a4s 982	. . . . 5 ⊢ (∀z z = x → ((z = x → z = y) ↔ (x = x → x = y)))
5	4	dral2 1153	. . . 4 ⊢ (∀z z = x → (∀z(z = x → z = y) ↔ ∀z(x = x → x = y)))
6		equid 1124	. . . . . . 7 ⊢ x = x
7	6	a1bi 197	. . . . . 6 ⊢ (x = y ↔ (x = x → x = y))
8	7	biimpr 152	. . . . 5 ⊢ ((x = x → x = y) → x = y)
9	8	a4s 982	. . . 4 ⊢ (∀z(x = x → x = y) → x = y)
10	5, 9	syl6bi 214	. . 3 ⊢ (∀z z = x → (∀z(z = x → z = y) → x = y))
11	10	a1d 12	. 2 ⊢ (∀z z = x → (¬ ∀z z = y → (∀z(z = x → z = y) → x = y)))
12		hbn1 1013	. . . . . . 7 ⊢ (¬ ∀z z = x → ∀z ¬ ∀z z = x)
13		hbn1 1013	. . . . . . 7 ⊢ (¬ ∀z z = y → ∀z ¬ ∀z z = y)
14	12, 13	hban 1007	. . . . . 6 ⊢ ((¬ ∀z z = x ⋀ ¬ ∀z z = y) → ∀z(¬ ∀z z = x ⋀ ¬ ∀z z = y))
15	6	hbth 999	. . . . . . . 8 ⊢ (x = x → ∀z x = x)
16	15	a1i 8	. . . . . . 7 ⊢ ((¬ ∀z z = x ⋀ ¬ ∀z z = y) → (x = x → ∀z x = x))
17		ax-12 966	. . . . . . . 8 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
18	17	imp 350	. . . . . . 7 ⊢ ((¬ ∀z z = x ⋀ ¬ ∀z z = y) → (x = y → ∀z x = y))
19	14, 16, 18	hbimd 1108	. . . . . 6 ⊢ ((¬ ∀z z = x ⋀ ¬ ∀z z = y) → ((x = x → x = y) → ∀z(x = x → x = y)))
20	14, 19	19.21ai 996	. . . . 5 ⊢ ((¬ ∀z z = x ⋀ ¬ ∀z z = y) → ∀z((x = x → x = y) → ∀z(x = x → x = y)))
21		equtr 1129	. . . . . . . 8 ⊢ (z = x → (x = x → z = x))
22		ax-8 962	. . . . . . . 8 ⊢ (z = x → (z = y → x = y))
23	21, 22	imim12d 29	. . . . . . 7 ⊢ (z = x → ((z = x → z = y) → (x = x → x = y)))
24	23	ax-gen 961	. . . . . 6 ⊢ ∀z(z = x → ((z = x → z = y) → (x = x → x = y)))
25		19.26 1065	. . . . . . 7 ⊢ (∀z(((x = x → x = y) → ∀z(x = x → x = y)) ⋀ (z = x → ((z = x → z = y) → (x = x → x = y)))) ↔ (∀z((x = x → x = y) → ∀z(x = x → x = y)) ⋀ ∀z(z = x → ((z = x → z = y) → (x = x → x = y)))))
26		a4imt 1156	. . . . . . 7 ⊢ (∀z(((x = x → x = y) → ∀z(x = x → x = y)) ⋀ (z = x → ((z = x → z = y) → (x = x → x = y)))) → (∀z(z = x → z = y) → (x = x → x = y)))
27	25, 26	sylbir 201	. . . . . 6 ⊢ ((∀z((x = x → x = y) → ∀z(x = x → x = y)) ⋀ ∀z(z = x → ((z = x → z = y) → (x = x → x = y)))) → (∀z(z = x → z = y) → (x = x → x = y)))
28	24, 27	mpan2 695	. . . . 5 ⊢ (∀z((x = x → x = y) → ∀z(x = x → x = y)) → (∀z(z = x → z = y) → (x = x → x = y)))
29	20, 28	syl 10	. . . 4 ⊢ ((¬ ∀z z = x ⋀ ¬ ∀z z = y) → (∀z(z = x → z = y) → (x = x → x = y)))
30	6, 29	mpii 45	. . 3 ⊢ ((¬ ∀z z = x ⋀ ¬ ∀z z = y) → (∀z(z = x → z = y) → x = y))
31	30	ex 373	. 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (∀z(z = x → z = y) → x = y)))
32	11, 31	pm2.61i 126	1 ⊢ (¬ ∀z z = y → (∀z(z = x → z = y) → x = y))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138

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

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