Theorem copsexg 1902

Description: Substitution of class A for ordered pair <.x, y>..

Assertion

Ref	Expression
copsexg	|- (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph)))

Distinct variable group(s):   x,y,A

Proof of Theorem copsexg

Step	Hyp	Ref	Expression
1		visset 1350	. . . 4 |- x e. V
2		visset 1350	. . . 4 |- y e. V
3	1, 2	eqvinop 1901	. . 3 |- (A = <.x, y>. <-> E.zE.w(A = <.z, w>. /\ <.z, w>. = <.x, y>.))
4		cleqcom 1103	. . . . . . . . 9 |- (<.z, w>. = <.x, y>. <-> <.x, y>. = <.z, w>.)
5		visset 1350	. . . . . . . . . 10 |- w e. V
6	1, 2, 5	opth 1898	. . . . . . . . 9 |- (<.x, y>. = <.z, w>. <-> (x = z /\ y = w))
7	4, 6	bitr 151	. . . . . . . 8 |- (<.z, w>. = <.x, y>. <-> (x = z /\ y = w))
8		ceqex 1410	. . . . . . . . 9 |- (y = w -> (ph <-> E.y(y = w /\ ph)))
9		ceqex 1410	. . . . . . . . 9 |- (x = z -> (E.y(y = w /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph))))
10	8, 9	sylan9bbr 419	. . . . . . . 8 |- ((x = z /\ y = w) -> (ph <-> E.x(x = z /\ E.y(y = w /\ ph))))
11	7, 10	sylbi 174	. . . . . . 7 |- (<.z, w>. = <.x, y>. -> (ph <-> E.x(x = z /\ E.y(y = w /\ ph))))
12	7	anbi1i 368	. . . . . . . . . . 11 |- ((<.z, w>. = <.x, y>. /\ ph) <-> ((x = z /\ y = w) /\ ph))
13		anass 336	. . . . . . . . . . 11 |- (((x = z /\ y = w) /\ ph) <-> (x = z /\ (y = w /\ ph)))
14	12, 13	bitr 151	. . . . . . . . . 10 |- ((<.z, w>. = <.x, y>. /\ ph) <-> (x = z /\ (y = w /\ ph)))
15	14	biex 733	. . . . . . . . 9 |- (E.y(<.z, w>. = <.x, y>. /\ ph) <-> E.y(x = z /\ (y = w /\ ph)))
16		19.42v 966	. . . . . . . . 9 |- (E.y(x = z /\ (y = w /\ ph)) <-> (x = z /\ E.y(y = w /\ ph)))
17	15, 16	bitr 151	. . . . . . . 8 |- (E.y(<.z, w>. = <.x, y>. /\ ph) <-> (x = z /\ E.y(y = w /\ ph)))
18	17	biex 733	. . . . . . 7 |- (E.xE.y(<.z, w>. = <.x, y>. /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph)))
19	11, 18	syl6bbr 416	. . . . . 6 |- (<.z, w>. = <.x, y>. -> (ph <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
20		cleq1 1107	. . . . . . 7 |- (A = <.z, w>. -> (A = <.x, y>. <-> <.z, w>. = <.x, y>.))
21	20	anbi1d 469	. . . . . . . . 9 |- (A = <.z, w>. -> ((A = <.x, y>. /\ ph) <-> (<.z, w>. = <.x, y>. /\ ph)))
22	21	bi2exdv 938	. . . . . . . 8 |- (A = <.z, w>. -> (E.xE.y(A = <.x, y>. /\ ph) <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
23	22	bibi2d 470	. . . . . . 7 |- (A = <.z, w>. -> ((ph <-> E.xE.y(A = <.x, y>. /\ ph)) <-> (ph <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph))))
24	20, 23	imbi12d 474	. . . . . 6 |- (A = <.z, w>. -> ((A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))) <-> (<.z, w>. = <.x, y>. -> (ph <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))))
25	19, 24	mpbiri 169	. . . . 5 |- (A = <.z, w>. -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
26	25	adantr 306	. . . 4 |- ((A = <.z, w>. /\ <.z, w>. = <.x, y>.) -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
27	26	19.23aivv 953	. . 3 |- (E.zE.w(A = <.z, w>. /\ <.z, w>. = <.x, y>.) -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
28	3, 27	sylbi 174	. 2 |- (A = <.x, y>. -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
29	28	pm2.43i 58	1 |- (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = weq 797   = wceq 1091  <.cop 1810

This theorem is referenced by:  copsex2g 1903  mosubop 1911  ssopab2 2119

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

metamath.org