Theorem equveli 1639

Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1638.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)

Assertion

Ref	Expression
equveli	|- (A.z(z = x <-> z = y) -> x = y)

Proof of Theorem equveli

Step	Hyp	Ref	Expression
1		albiim 1373	. 2 |- (A.z(z = x <-> z = y) <-> (A.z(z = x -> z = y) /\ A.z(z = y -> z = x)))
2		ax12or 1400	. . 3 |- (A.z z = x \/ (A.z z = y \/ A.z(x = y -> A.z x = y)))
3		equequ1 1595	. . . . . . . . 9 |- (z = x -> (z = x <-> x = x))
4		equequ1 1595	. . . . . . . . 9 |- (z = x -> (z = y <-> x = y))
5	3, 4	imbi12d 223	. . . . . . . 8 |- (z = x -> ((z = x -> z = y) <-> (x = x -> x = y)))
6	5	sps 1427	. . . . . . 7 |- (A.z z = x -> ((z = x -> z = y) <-> (x = x -> x = y)))
7	6	dral2 1616	. . . . . 6 |- (A.z z = x -> (A.z(z = x -> z = y) <-> A.z(x = x -> x = y)))
8		equid 1586	. . . . . . . . 9 |- x = x
9	8	a1bi 232	. . . . . . . 8 |- (x = y <-> (x = x -> x = y))
10	9	biimpri 124	. . . . . . 7 |- ((x = x -> x = y) -> x = y)
11	10	sps 1427	. . . . . 6 |- (A.z(x = x -> x = y) -> x = y)
12	7, 11	syl6bi 152	. . . . 5 |- (A.z z = x -> (A.z(z = x -> z = y) -> x = y))
13	12	adantrd 264	. . . 4 |- (A.z z = x -> ((A.z(z = x -> z = y) /\ A.z(z = y -> z = x)) -> x = y))
14		equequ1 1595	. . . . . . . . . 10 |- (z = y -> (z = y <-> y = y))
15		equequ1 1595	. . . . . . . . . 10 |- (z = y -> (z = x <-> y = x))
16	14, 15	imbi12d 223	. . . . . . . . 9 |- (z = y -> ((z = y -> z = x) <-> (y = y -> y = x)))
17	16	sps 1427	. . . . . . . 8 |- (A.z z = y -> ((z = y -> z = x) <-> (y = y -> y = x)))
18	17	dral1 1615	. . . . . . 7 |- (A.z z = y -> (A.z(z = y -> z = x) <-> A.y(y = y -> y = x)))
19		equid 1586	. . . . . . . . 9 |- y = y
20		ax-4 1397	. . . . . . . . 9 |- (A.y(y = y -> y = x) -> (y = y -> y = x))
21	19, 20	mpi 15	. . . . . . . 8 |- (A.y(y = y -> y = x) -> y = x)
22		equcomi 1589	. . . . . . . 8 |- (y = x -> x = y)
23	21, 22	syl 14	. . . . . . 7 |- (A.y(y = y -> y = x) -> x = y)
24	18, 23	syl6bi 152	. . . . . 6 |- (A.z z = y -> (A.z(z = y -> z = x) -> x = y))
25	24	adantld 263	. . . . 5 |- (A.z z = y -> ((A.z(z = x -> z = y) /\ A.z(z = y -> z = x)) -> x = y))
26		hba1 1430	. . . . . . . . . 10 |- (A.z(x = y -> A.z x = y) -> A.zA.z(x = y -> A.z x = y))
27		hbequid 1403	. . . . . . . . . . 11 |- (x = x -> A.z x = x)
28	27	a1i 9	. . . . . . . . . 10 |- (A.z(x = y -> A.z x = y) -> (x = x -> A.z x = x))
29		ax-4 1397	. . . . . . . . . 10 |- (A.z(x = y -> A.z x = y) -> (x = y -> A.z x = y))
30	26, 28, 29	hbimd 1462	. . . . . . . . 9 |- (A.z(x = y -> A.z x = y) -> ((x = x -> x = y) -> A.z(x = x -> x = y)))
31	30	a5i 1432	. . . . . . . 8 |- (A.z(x = y -> A.z x = y) -> A.z((x = x -> x = y) -> A.z(x = x -> x = y)))
32		equtr 1592	. . . . . . . . . 10 |- (z = x -> (x = x -> z = x))
33		ax-8 1392	. . . . . . . . . 10 |- (z = x -> (z = y -> x = y))
34	32, 33	imim12d 68	. . . . . . . . 9 |- (z = x -> ((z = x -> z = y) -> (x = x -> x = y)))
35	34	ax-gen 1335	. . . . . . . 8 |- A.z(z = x -> ((z = x -> z = y) -> (x = x -> x = y)))
36		19.26 1367	. . . . . . . . 9 |- (A.z(((x = x -> x = y) -> A.z(x = x -> x = y)) /\ (z = x -> ((z = x -> z = y) -> (x = x -> x = y)))) <-> (A.z((x = x -> x = y) -> A.z(x = x -> x = y)) /\ A.z(z = x -> ((z = x -> z = y) -> (x = x -> x = y)))))
37		spimth 1620	. . . . . . . . 9 |- (A.z(((x = x -> x = y) -> A.z(x = x -> x = y)) /\ (z = x -> ((z = x -> z = y) -> (x = x -> x = y)))) -> (A.z(z = x -> z = y) -> (x = x -> x = y)))
38	36, 37	sylbir 125	. . . . . . . 8 |- ((A.z((x = x -> x = y) -> A.z(x = x -> x = y)) /\ A.z(z = x -> ((z = x -> z = y) -> (x = x -> x = y)))) -> (A.z(z = x -> z = y) -> (x = x -> x = y)))
39	31, 35, 38	sylancl 392	. . . . . . 7 |- (A.z(x = y -> A.z x = y) -> (A.z(z = x -> z = y) -> (x = x -> x = y)))
40	8, 39	mpii 39	. . . . . 6 |- (A.z(x = y -> A.z x = y) -> (A.z(z = x -> z = y) -> x = y))
41	40	adantrd 264	. . . . 5 |- (A.z(x = y -> A.z x = y) -> ((A.z(z = x -> z = y) /\ A.z(z = y -> z = x)) -> x = y))
42	25, 41	jaoi 635	. . . 4 |- ((A.z z = y \/ A.z(x = y -> A.z x = y)) -> ((A.z(z = x -> z = y) /\ A.z(z = y -> z = x)) -> x = y))
43	13, 42	jaoi 635	. . 3 |- ((A.z z = x \/ (A.z z = y \/ A.z(x = y -> A.z x = y))) -> ((A.z(z = x -> z = y) /\ A.z(z = y -> z = x)) -> x = y))
44	2, 43	ax-mp 7	. 2 |- ((A.z(z = x -> z = y) /\ A.z(z = y -> z = x)) -> x = y)
45	1, 44	sylbi 114	1 |- (A.z(z = x <-> z = y) -> x = y)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  A.wal 1240   = wceq 1242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)