Theorem equveli 1988

Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1987.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)

Assertion

Ref	Expression
equveli	⊢ (∀z(z = x ↔ z = y) → x = y)

Proof of Theorem equveli

Step	Hyp	Ref	Expression
1		albiim 1611	. 2 ⊢ (∀z(z = x ↔ z = y) ↔ (∀z(z = x → z = y) ∧ ∀z(z = y → z = x)))
2		equequ1 1684	. . . . . . . 8 ⊢ (z = y → (z = y ↔ y = y))
3		equequ1 1684	. . . . . . . 8 ⊢ (z = y → (z = x ↔ y = x))
4	2, 3	imbi12d 311	. . . . . . 7 ⊢ (z = y → ((z = y → z = x) ↔ (y = y → y = x)))
5	4	sps 1754	. . . . . 6 ⊢ (∀z z = y → ((z = y → z = x) ↔ (y = y → y = x)))
6	5	dral1 1965	. . . . 5 ⊢ (∀z z = y → (∀z(z = y → z = x) ↔ ∀y(y = y → y = x)))
7		equid 1676	. . . . . . 7 ⊢ y = y
8		sp 1747	. . . . . . 7 ⊢ (∀y(y = y → y = x) → (y = y → y = x))
9	7, 8	mpi 16	. . . . . 6 ⊢ (∀y(y = y → y = x) → y = x)
10		equcomi 1679	. . . . . 6 ⊢ (y = x → x = y)
11	9, 10	syl 15	. . . . 5 ⊢ (∀y(y = y → y = x) → x = y)
12	6, 11	syl6bi 219	. . . 4 ⊢ (∀z z = y → (∀z(z = y → z = x) → x = y))
13	12	adantld 453	. . 3 ⊢ (∀z z = y → ((∀z(z = x → z = y) ∧ ∀z(z = y → z = x)) → x = y))
14		equequ1 1684	. . . . . . . . . 10 ⊢ (z = x → (z = x ↔ x = x))
15		equequ1 1684	. . . . . . . . . 10 ⊢ (z = x → (z = y ↔ x = y))
16	14, 15	imbi12d 311	. . . . . . . . 9 ⊢ (z = x → ((z = x → z = y) ↔ (x = x → x = y)))
17	16	sps 1754	. . . . . . . 8 ⊢ (∀z z = x → ((z = x → z = y) ↔ (x = x → x = y)))
18	17	dral2 1966	. . . . . . 7 ⊢ (∀z z = x → (∀z(z = x → z = y) ↔ ∀z(x = x → x = y)))
19		equid 1676	. . . . . . . . . 10 ⊢ x = x
20	19	a1bi 327	. . . . . . . . 9 ⊢ (x = y ↔ (x = x → x = y))
21	20	biimpri 197	. . . . . . . 8 ⊢ ((x = x → x = y) → x = y)
22	21	sps 1754	. . . . . . 7 ⊢ (∀z(x = x → x = y) → x = y)
23	18, 22	syl6bi 219	. . . . . 6 ⊢ (∀z z = x → (∀z(z = x → z = y) → x = y))
24	23	a1d 22	. . . . 5 ⊢ (∀z z = x → (¬ ∀z z = y → (∀z(z = x → z = y) → x = y)))
25		nfeqf 1958	. . . . . . 7 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → Ⅎz x = y)
26		equtr 1682	. . . . . . . . . 10 ⊢ (z = x → (x = x → z = x))
27		ax-8 1675	. . . . . . . . . 10 ⊢ (z = x → (z = y → x = y))
28	26, 27	imim12d 68	. . . . . . . . 9 ⊢ (z = x → ((z = x → z = y) → (x = x → x = y)))
29	19, 28	mpii 39	. . . . . . . 8 ⊢ (z = x → ((z = x → z = y) → x = y))
30	29	ax-gen 1546	. . . . . . 7 ⊢ ∀z(z = x → ((z = x → z = y) → x = y))
31		spimt 1974	. . . . . . 7 ⊢ ((Ⅎz x = y ∧ ∀z(z = x → ((z = x → z = y) → x = y))) → (∀z(z = x → z = y) → x = y))
32	25, 30, 31	sylancl 643	. . . . . 6 ⊢ ((¬ ∀z z = x ∧ ¬ ∀z z = y) → (∀z(z = x → z = y) → x = y))
33	32	ex 423	. . . . 5 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (∀z(z = x → z = y) → x = y)))
34	24, 33	pm2.61i 156	. . . 4 ⊢ (¬ ∀z z = y → (∀z(z = x → z = y) → x = y))
35	34	adantrd 454	. . 3 ⊢ (¬ ∀z z = y → ((∀z(z = x → z = y) ∧ ∀z(z = y → z = x)) → x = y))
36	13, 35	pm2.61i 156	. 2 ⊢ ((∀z(z = x → z = y) ∧ ∀z(z = y → z = x)) → x = y)
37	1, 36	sylbi 187	1 ⊢ (∀z(z = x ↔ z = y) → x = y)

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

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)