Theorem equveli 1773

Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1772.) (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 1501	. 2 |- ( A. z ( z = x <-> z = y ) <-> ( A. z ( z = x -> z = y ) /\ A. z ( z = y -> z = x ) ) )
2		ax12or 1522	. . 3 |- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
3		equequ1 1726	. . . . . . . . 9 |- ( z = x -> ( z = x <-> x = x ) )
4		equequ1 1726	. . . . . . . . 9 |- ( z = x -> ( z = y <-> x = y ) )
5	3, 4	imbi12d 234	. . . . . . . 8 |- ( z = x -> ( ( z = x -> z = y ) <-> ( x = x -> x = y ) ) )
6	5	sps 1551	. . . . . . 7 |- ( A. z z = x -> ( ( z = x -> z = y ) <-> ( x = x -> x = y ) ) )
7	6	dral2 1745	. . . . . 6 |- ( A. z z = x -> ( A. z ( z = x -> z = y ) <-> A. z ( x = x -> x = y ) ) )
8		equid 1715	. . . . . . . . 9 |- x = x
9	8	a1bi 243	. . . . . . . 8 |- ( x = y <-> ( x = x -> x = y ) )
10	9	biimpri 133	. . . . . . 7 |- ( ( x = x -> x = y ) -> x = y )
11	10	sps 1551	. . . . . 6 |- ( A. z ( x = x -> x = y ) -> x = y )
12	7, 11	biimtrdi 163	. . . . 5 |- ( A. z z = x -> ( A. z ( z = x -> z = y ) -> x = y ) )
13	12	adantrd 279	. . . 4 |- ( A. z z = x -> ( ( A. z ( z = x -> z = y ) /\ A. z ( z = y -> z = x ) ) -> x = y ) )
14		equequ1 1726	. . . . . . . . . 10 |- ( z = y -> ( z = y <-> y = y ) )
15		equequ1 1726	. . . . . . . . . 10 |- ( z = y -> ( z = x <-> y = x ) )
16	14, 15	imbi12d 234	. . . . . . . . 9 |- ( z = y -> ( ( z = y -> z = x ) <-> ( y = y -> y = x ) ) )
17	16	sps 1551	. . . . . . . 8 |- ( A. z z = y -> ( ( z = y -> z = x ) <-> ( y = y -> y = x ) ) )
18	17	dral1 1744	. . . . . . 7 |- ( A. z z = y -> ( A. z ( z = y -> z = x ) <-> A. y ( y = y -> y = x ) ) )
19		equid 1715	. . . . . . . . 9 |- y = y
20		ax-4 1524	. . . . . . . . 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 1718	. . . . . . . 8 |- ( y = x -> x = y )
23	21, 22	syl 14	. . . . . . 7 |- ( A. y ( y = y -> y = x ) -> x = y )
24	18, 23	biimtrdi 163	. . . . . 6 |- ( A. z z = y -> ( A. z ( z = y -> z = x ) -> x = y ) )
25	24	adantld 278	. . . . 5 |- ( A. z z = y -> ( ( A. z ( z = x -> z = y ) /\ A. z ( z = y -> z = x ) ) -> x = y ) )
26		hba1 1554	. . . . . . . . . 10 |- ( A. z ( x = y -> A. z x = y ) -> A. z A. z ( x = y -> A. z x = y ) )
27		hbequid 1527	. . . . . . . . . . 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 1524	. . . . . . . . . 10 |- ( A. z ( x = y -> A. z x = y ) -> ( x = y -> A. z x = y ) )
30	26, 28, 29	hbimd 1587	. . . . . . . . 9 |- ( A. z ( x = y -> A. z x = y ) -> ( ( x = x -> x = y ) -> A. z ( x = x -> x = y ) ) )
31	30	a5i 1557	. . . . . . . 8 |- ( A. z ( x = y -> A. z x = y ) -> A. z ( ( x = x -> x = y ) -> A. z ( x = x -> x = y ) ) )
32		equtr 1723	. . . . . . . . . 10 |- ( z = x -> ( x = x -> z = x ) )
33		ax-8 1518	. . . . . . . . . 10 |- ( z = x -> ( z = y -> x = y ) )
34	32, 33	imim12d 74	. . . . . . . . 9 |- ( z = x -> ( ( z = x -> z = y ) -> ( x = x -> x = y ) ) )
35	34	ax-gen 1463	. . . . . . . 8 |- A. z ( z = x -> ( ( z = x -> z = y ) -> ( x = x -> x = y ) ) )
36		19.26 1495	. . . . . . . . 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 1749	. . . . . . . . 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 135	. . . . . . . 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 413	. . . . . . 7 |- ( A. z ( x = y -> A. z x = y ) -> ( A. z ( z = x -> z = y ) -> ( x = x -> x = y ) ) )
40	8, 39	mpii 44	. . . . . 6 |- ( A. z ( x = y -> A. z x = y ) -> ( A. z ( z = x -> z = y ) -> x = y ) )
41	40	adantrd 279	. . . . 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 717	. . . 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 717	. . 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 5	. 2 |- ( ( A. z ( z = x -> z = y ) /\ A. z ( z = y -> z = x ) ) -> x = y )
45	1, 44	sylbi 121	1 |- ( A. z ( z = x <-> z = y ) -> x = y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)