Theorem copsexgOLD 3538

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

Assertion

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

Distinct variable group:   x,A,y

Proof of Theorem copsexgOLD

Step	Hyp	Ref	Expression
1		visset 2295	. . . 4 |- x e. _V
2		visset 2295	. . . 4 |- y e. _V
3	1, 2	eqvinop 3536	. . 3 |- (A = <.x, y>. <-> E.zE.w(A = <.z, w>. /\ <.z, w>. = <.x, y>.))
4		eqcom 1886	. . . . . . . . 9 |- (<.z, w>. = <.x, y>. <-> <.x, y>. = <.z, w>.)
5		visset 2295	. . . . . . . . . 10 |- w e. _V
6	1, 2, 5	opth 3532	. . . . . . . . 9 |- (<.x, y>. = <.z, w>. <-> (x = z /\ y = w))
7	4, 6	bitri 190	. . . . . . . 8 |- (<.z, w>. = <.x, y>. <-> (x = z /\ y = w))
8		ceqex 2391	. . . . . . . . 9 |- (y = w -> (ph <-> E.y(y = w /\ ph)))
9		ceqex 2391	. . . . . . . . 9 |- (x = z -> (E.y(y = w /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph))))
10	8, 9	sylan9bbr 600	. . . . . . . 8 |- ((x = z /\ y = w) -> (ph <-> E.x(x = z /\ E.y(y = w /\ ph))))
11	7, 10	sylbi 216	. . . . . . 7 |- (<.z, w>. = <.x, y>. -> (ph <-> E.x(x = z /\ E.y(y = w /\ ph))))
12	7	anbi1i 539	. . . . . . . . . . 11 |- ((<.z, w>. = <.x, y>. /\ ph) <-> ((x = z /\ y = w) /\ ph))
13		anass 487	. . . . . . . . . . 11 |- (((x = z /\ y = w) /\ ph) <-> (x = z /\ (y = w /\ ph)))
14	12, 13	bitri 190	. . . . . . . . . 10 |- ((<.z, w>. = <.x, y>. /\ ph) <-> (x = z /\ (y = w /\ ph)))
15	14	exbii 1398	. . . . . . . . 9 |- (E.y(<.z, w>. = <.x, y>. /\ ph) <-> E.y(x = z /\ (y = w /\ ph)))
16		19.42v 1688	. . . . . . . . 9 |- (E.y(x = z /\ (y = w /\ ph)) <-> (x = z /\ E.y(y = w /\ ph)))
17	15, 16	bitri 190	. . . . . . . 8 |- (E.y(<.z, w>. = <.x, y>. /\ ph) <-> (x = z /\ E.y(y = w /\ ph)))
18	17	exbii 1398	. . . . . . 7 |- (E.xE.y(<.z, w>. = <.x, y>. /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph)))
19	11, 18	syl6bbr 597	. . . . . 6 |- (<.z, w>. = <.x, y>. -> (ph <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
20		eqeq1 1890	. . . . . . 7 |- (A = <.z, w>. -> (A = <.x, y>. <-> <.z, w>. = <.x, y>.))
21	20	anbi1d 679	. . . . . . . . 9 |- (A = <.z, w>. -> ((A = <.x, y>. /\ ph) <-> (<.z, w>. = <.x, y>. /\ ph)))
22	21	2exbidv 1659	. . . . . . . 8 |- (A = <.z, w>. -> (E.xE.y(A = <.x, y>. /\ ph) <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
23	22	bibi2d 680	. . . . . . 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 688	. . . . . 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 211	. . . . 5 |- (A = <.z, w>. -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
26	25	adantr 425	. . . 4 |- ((A = <.z, w>. /\ <.z, w>. = <.x, y>.) -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
27	26	19.23aivv 1675	. . 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 216	. 2 |- (A = <.x, y>. -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
29	28	pm2.43i 78	1 |- (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298  E.wex 1326  <.cop 3046

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053

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