Theorem copsexg 2787

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 1809	. . . 4 |- x e. V
2		visset 1809	. . . 4 |- y e. V
3	1, 2	eqvinop 2786	. . 3 |- (A = <.x, y>. <-> E.zE.w(A = <.z, w>. /\ <.z, w>. = <.x, y>.))
4		eqcom 1474	. . . . . . . . 9 |- (<.z, w>. = <.x, y>. <-> <.x, y>. = <.z, w>.)
5		visset 1809	. . . . . . . . . 10 |- w e. V
6	1, 2, 5	opth 2782	. . . . . . . . 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 1882	. . . . . . . . 9 |- (y = w -> (ph <-> E.y(y = w /\ ph)))
9		ceqex 1882	. . . . . . . . 9 |- (x = z -> (E.y(y = w /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph))))
10	8, 9	sylan9bbr 540	. . . . . . . 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 481	. . . . . . . . . . 11 |- ((<.z, w>. = <.x, y>. /\ ph) <-> ((x = z /\ y = w) /\ ph))
13		anass 439	. . . . . . . . . . 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 1049	. . . . . . . . 9 |- (E.y(<.z, w>. = <.x, y>. /\ ph) <-> E.y(x = z /\ (y = w /\ ph)))
16		19.42v 1306	. . . . . . . . 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 1049	. . . . . . 7 |- (E.xE.y(<.z, w>. = <.x, y>. /\ ph) <-> E.x(x = z /\ E.y(y = w /\ ph)))
19	11, 18	syl6bbr 537	. . . . . 6 |- (<.z, w>. = <.x, y>. -> (ph <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
20		eqeq1 1478	. . . . . . 7 |- (A = <.z, w>. -> (A = <.x, y>. <-> <.z, w>. = <.x, y>.))
21	20	anbi1d 616	. . . . . . . . 9 |- (A = <.z, w>. -> ((A = <.x, y>. /\ ph) <-> (<.z, w>. = <.x, y>. /\ ph)))
22	21	2exbidv 1279	. . . . . . . 8 |- (A = <.z, w>. -> (E.xE.y(A = <.x, y>. /\ ph) <-> E.xE.y(<.z, w>. = <.x, y>. /\ ph)))
23	22	bibi2d 617	. . . . . . 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 625	. . . . . 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 389	. . . 4 |- ((A = <.z, w>. /\ <.z, w>. = <.x, y>.) -> (A = <.x, y>. -> (ph <-> E.xE.y(A = <.x, y>. /\ ph))))
27	26	19.23aivv 1294	. . 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 954  E.wex 978  <.cop 2407

This theorem is referenced by:  copsex2g 2788  mosubopt 2799  ssopab2 2817

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412

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