Theorem copsexg 2798

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:   x,y,A

Proof of Theorem copsexg

Step	Hyp	Ref	Expression
1		visset 1816	. . . 4 |- x e. V
2		visset 1816	. . . 4 |- y e. V
3	1, 2	eqvinop 2797	. . 3 |- (A = <.x, y>. <-> E.zE.w(A = <.z, w>. /\ <.z, w>. = <.x, y>.))
4		eqcom 1480	. . . . . . . . 9 |- (<.z, w>. = <.x, y>. <-> <.x, y>. = <.z, w>.)
5		visset 1816	. . . . . . . . . 10 |- w e. V
6	1, 2, 5	opth 2793	. . . . . . . . 9 |- (<.x, y>. = <.z, w>. <-> (x = z /\ y = w))
7	4, 6	bitr 173	. . . . . . . 8 |- (<.z, w>. = <.x, y>. <-> (x = z /\ y = w))
8		ceqex 1889	. . . . . . . . 9 |- (y = w -> (ph <-> E.y(y = w /\ ph)))
9		ceqex 1889	. . . . . . . . 9 |- (x = z -> (E.y(y = w /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph))))
10	8, 9	sylan9bbr 543	. . . . . . . 8 |- ((x = z /\ y = w) -> (ph <-> E.x(x = z /\ E.y(y = w /\ ph))))
11	7, 10	sylbi 199	. . . . . . 7 |- (<.z, w>. = <.x, y>. -> (ph <-> E.x(x = z /\ E.y(y = w /\ ph))))
12	7	anbi1i 483	. . . . . . . . . . 11 |- ((<.z, w>. = <.x, y>. /\ ph) <-> ((x = z /\ y = w) /\ ph))
13		anass 441	. . . . . . . . . . 11 |- (((x = z /\ y = w) /\ ph) <-> (x = z /\ (y = w /\ ph)))
14	12, 13	bitr 173	. . . . . . . . . 10 |- ((<.z, w>. = <.x, y>. /\ ph) <-> (x = z /\ (y = w /\ ph)))
15	14	exbii 1053	. . . . . . . . 9 |- (E.y(<.z, w>. = <.x, y>. /\ ph) <-> E.y(x = z /\ (y = w /\ ph)))
16		19.42v 1310	. . . . . . . . 9 |- (E.y(x = z /\ (y = w /\ ph)) <-> (x = z /\ E.y(y = w /\ ph)))
17	15, 16	bitr 173	. . . . . . . 8 |- (E.y(<.z, w>. = <.x, y>. /\ ph) <-> (x = z /\ E.y(y = w /\ ph)))
18	17	exbii 1053	. . . . . . 7 |- (E.xE.y(<.z, w>. = <.x, y>. /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph)))
19	11, 18	syl6bbr 540	. . . . . 6 |- (<.z, w>. = <.x, y>. -> (ph <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
20		eqeq1 1484	. . . . . . 7 |- (A = <.z, w>. -> (A = <.x, y>. <-> <.z, w>. = <.x, y>.))
21	20	anbi1d 619	. . . . . . . . 9 |- (A = <.z, w>. -> ((A = <.x, y>. /\ ph) <-> (<.z, w>. = <.x, y>. /\ ph)))
22	21	2exbidv 1283	. . . . . . . 8 |- (A = <.z, w>. -> (E.xE.y(A = <.x, y>. /\ ph) <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
23	22	bibi2d 620	. . . . . . 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 628	. . . . . 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 194	. . . . 5 |- (A = <.z, w>. -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
26	25	adantr 391	. . . 4 |- ((A = <.z, w>. /\ <.z, w>. = <.x, y>.) -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
27	26	19.23aivv 1298	. . 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 199	. 2 |- (A = <.x, y>. -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
29	28	pm2.43i 64	1 |- (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958  E.wex 982  <.cop 2415

This theorem is referenced by:  copsex2g 2799  mosubopt 2810  ssopab2 2828

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420

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