Theorem 2mo 1447

Description: Two equivalent expressions for double "at most one."

Assertion

Ref	Expression
2mo	|- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))

Distinct variable groups:   x,y,z,w   ph,z,w

Proof of Theorem 2mo

Step	Hyp	Ref	Expression
1		equequ2 1135	. . . . . . 7 |- (v = z -> (x = v <-> x = z))
2		equequ2 1135	. . . . . . 7 |- (u = w -> (y = u <-> y = w))
3	1, 2	bi2anan9 632	. . . . . 6 |- ((v = z /\ u = w) -> ((x = v /\ y = u) <-> (x = z /\ y = w)))
4	3	imbi2d 612	. . . . 5 |- ((v = z /\ u = w) -> ((ph -> (x = v /\ y = u)) <-> (ph -> (x = z /\ y = w))))
5	4	2albidv 1280	. . . 4 |- ((v = z /\ u = w) -> (A.xA.y(ph -> (x = v /\ y = u)) <-> A.xA.y(ph -> (x = z /\ y = w))))
6	5	cbvex2v 1319	. . 3 |- (E.vE.uA.xA.y(ph -> (x = v /\ y = u)) <-> E.zE.wA.xA.y(ph -> (x = z /\ y = w)))
7		ax-17 971	. . . . . . . . 9 |- ((ph -> (x = v /\ y = u)) -> A.z(ph -> (x = v /\ y = u)))
8		ax-17 971	. . . . . . . . 9 |- ((ph -> (x = v /\ y = u)) -> A.w(ph -> (x = v /\ y = u)))
9		hbs1 1332	. . . . . . . . . 10 |- ([z / x][w / y]ph -> A.x[z / x][w / y]ph)
10		ax-17 971	. . . . . . . . . 10 |- ((z = v /\ w = u) -> A.x(z = v /\ w = u))
11	9, 10	hbim 1007	. . . . . . . . 9 |- (([z / x][w / y]ph -> (z = v /\ w = u)) -> A.x([z / x][w / y]ph -> (z = v /\ w = u)))
12		hbs1 1332	. . . . . . . . . . 11 |- ([w / y]ph -> A.y[w / y]ph)
13	12	hbsb 1333	. . . . . . . . . 10 |- ([z / x][w / y]ph -> A.y[z / x][w / y]ph)
14		ax-17 971	. . . . . . . . . 10 |- ((z = v /\ w = u) -> A.y(z = v /\ w = u))
15	13, 14	hbim 1007	. . . . . . . . 9 |- (([z / x][w / y]ph -> (z = v /\ w = u)) -> A.y([z / x][w / y]ph -> (z = v /\ w = u)))
16		sbequ12 1181	. . . . . . . . . . 11 |- (y = w -> (ph <-> [w / y]ph))
17		sbequ12 1181	. . . . . . . . . . 11 |- (x = z -> ([w / y]ph <-> [z / x][w / y]ph))
18	16, 17	sylan9bbr 541	. . . . . . . . . 10 |- ((x = z /\ y = w) -> (ph <-> [z / x][w / y]ph))
19		equequ1 1134	. . . . . . . . . . 11 |- (x = z -> (x = v <-> z = v))
20		equequ1 1134	. . . . . . . . . . 11 |- (y = w -> (y = u <-> w = u))
21	19, 20	bi2anan9 632	. . . . . . . . . 10 |- ((x = z /\ y = w) -> ((x = v /\ y = u) <-> (z = v /\ w = u)))
22	18, 21	imbi12d 626	. . . . . . . . 9 |- ((x = z /\ y = w) -> ((ph -> (x = v /\ y = u)) <-> ([z / x][w / y]ph -> (z = v /\ w = u))))
23	7, 8, 11, 15, 22	cbval2 1316	. . . . . . . 8 |- (A.xA.y(ph -> (x = v /\ y = u)) <-> A.zA.w([z / x][w / y]ph -> (z = v /\ w = u)))
24	23	biimp 151	. . . . . . 7 |- (A.xA.y(ph -> (x = v /\ y = u)) -> A.zA.w([z / x][w / y]ph -> (z = v /\ w = u)))
25	24	ancli 296	. . . . . 6 |- (A.xA.y(ph -> (x = v /\ y = u)) -> (A.xA.y(ph -> (x = v /\ y = u)) /\ A.zA.w([z / x][w / y]ph -> (z = v /\ w = u))))
26		alcom 1032	. . . . . . . . 9 |- (A.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.zA.yA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))))
27	8, 15	aaan 1119	. . . . . . . . . 10 |- (A.yA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> (A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
28	27	albii 999	. . . . . . . . 9 |- (A.zA.yA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
29	26, 28	bitr 173	. . . . . . . 8 |- (A.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
30	29	albii 999	. . . . . . 7 |- (A.xA.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> A.xA.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))))
31		ax-17 971	. . . . . . . 8 |- (A.y(ph -> (x = v /\ y = u)) -> A.zA.y(ph -> (x = v /\ y = u)))
32	11	hbal 1005	. . . . . . . 8 |- (A.w([z / x][w / y]ph -> (z = v /\ w = u)) -> A.xA.w([z / x][w / y]ph -> (z = v /\ w = u)))
33	31, 32	aaan 1119	. . . . . . 7 |- (A.xA.z(A.y(ph -> (x = v /\ y = u)) /\ A.w([z / x][w / y]ph -> (z = v /\ w = u))) <-> (A.xA.y(ph -> (x = v /\ y = u)) /\ A.zA.w([z / x][w / y]ph -> (z = v /\ w = u))))
34	30, 33	bitr 173	. . . . . 6 |- (A.xA.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) <-> (A.xA.y(ph -> (x = v /\ y = u)) /\ A.zA.w([z / x][w / y]ph -> (z = v /\ w = u))))
35	25, 34	sylibr 200	. . . . 5 |- (A.xA.y(ph -> (x = v /\ y = u)) -> A.xA.yA.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))))
36		prth 556	. . . . . . . 8 |- (((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) -> ((ph /\ [z / x][w / y]ph) -> ((x = v /\ y = u) /\ (z = v /\ w = u))))
37		equtr2 1133	. . . . . . . . . 10 |- ((x = v /\ z = v) -> x = z)
38		equtr2 1133	. . . . . . . . . 10 |- ((y = u /\ w = u) -> y = w)
39	37, 38	anim12i 333	. . . . . . . . 9 |- (((x = v /\ z = v) /\ (y = u /\ w = u)) -> (x = z /\ y = w))
40	39	an4s 508	. . . . . . . 8 |- (((x = v /\ y = u) /\ (z = v /\ w = u)) -> (x = z /\ y = w))
41	36, 40	syl6 22	. . . . . . 7 |- (((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) -> ((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
42	41	19.20i2 993	. . . . . 6 |- (A.zA.w((ph -> (x = v /\ y = u)) /\ ([z / x][w / y]ph -> (z = v /\ w = u))) -> A.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
43	42	19.20i2 993	. . . . 5