Theorem copsex2t 4608

Description: Closed theorem form of copsex2g 4609. (Contributed by NM, 17-Feb-2013.)

Assertion

Ref	Expression
copsex2t	|- ((A.xA.y((x = A /\ y = B) -> (ph <-> ps)) /\ (A e. V /\ B e. W)) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))

Distinct variable groups:   x,y,ps   x,A,y   x,B,y

Allowed substitution hints:   ph(x,y)   V(x,y)   W(x,y)

Proof of Theorem copsex2t

Step	Hyp	Ref	Expression
1		elisset 2869	. . . 4 |- (A e. V -> E.x x = A)
2		elisset 2869	. . . 4 |- (B e. W -> E.y y = B)
3	1, 2	anim12i 549	. . 3 |- ((A e. V /\ B e. W) -> (E.x x = A /\ E.y y = B))
4		eeanv 1913	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
5	3, 4	sylibr 203	. 2 |- ((A e. V /\ B e. W) -> E.xE.y(x = A /\ y = B))
6		nfa1 1788	. . . 4 |- F/xA.xA.y((x = A /\ y = B) -> (ph <-> ps))
7		nfe1 1732	. . . . 5 |- F/xE.xE.y(<.A, B>. = <.x, y>. /\ ph)
8		nfv 1619	. . . . 5 |- F/xps
9	7, 8	nfbi 1834	. . . 4 |- F/x(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps)
10		nfa2 1855	. . . . 5 |- F/yA.xA.y((x = A /\ y = B) -> (ph <-> ps))
11		nfe1 1732	. . . . . . 7 |- F/yE.y(<.A, B>. = <.x, y>. /\ ph)
12	11	nfex 1843	. . . . . 6 |- F/yE.xE.y(<.A, B>. = <.x, y>. /\ ph)
13		nfv 1619	. . . . . 6 |- F/yps
14	12, 13	nfbi 1834	. . . . 5 |- F/y(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps)
15		opeq12 4580	. . . . . . . . 9 |- ((x = A /\ y = B) -> <.x, y>. = <.A, B>.)
16		copsexg 4607	. . . . . . . . . 10 |- (<.A, B>. = <.x, y>. -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
17	16	eqcoms 2356	. . . . . . . . 9 |- (<.x, y>. = <.A, B>. -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
18	15, 17	syl 15	. . . . . . . 8 |- ((x = A /\ y = B) -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
19	18	adantl 452	. . . . . . 7 |- ((A.xA.y((x = A /\ y = B) -> (ph <-> ps)) /\ (x = A /\ y = B)) -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
20		sp 1747	. . . . . . . . 9 |- (A.xA.y((x = A /\ y = B) -> (ph <-> ps)) -> A.y((x = A /\ y = B) -> (ph <-> ps)))
21	20	19.21bi 1758	. . . . . . . 8 |- (A.xA.y((x = A /\ y = B) -> (ph <-> ps)) -> ((x = A /\ y = B) -> (ph <-> ps)))
22	21	imp 418	. . . . . . 7 |- ((A.xA.y((x = A /\ y = B) -> (ph <-> ps)) /\ (x = A /\ y = B)) -> (ph <-> ps))
23	19, 22	bitr3d 246	. . . . . 6 |- ((A.xA.y((x = A /\ y = B) -> (ph <-> ps)) /\ (x = A /\ y = B)) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
24	23	ex 423	. . . . 5 |- (A.xA.y((x = A /\ y = B) -> (ph <-> ps)) -> ((x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps)))
25	10, 14, 24	exlimd 1806	. . . 4 |- (A.xA.y((x = A /\ y = B) -> (ph <-> ps)) -> (E.y(x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps)))
26	6, 9, 25	exlimd 1806	. . 3 |- (A.xA.y((x = A /\ y = B) -> (ph <-> ps)) -> (E.xE.y(x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps)))
27	26	imp 418	. 2 |- ((A.xA.y((x = A /\ y = B) -> (ph <-> ps)) /\ E.xE.y(x = A /\ y = B)) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
28	5, 27	sylan2 460	1 |- ((A.xA.y((x = A /\ y = B) -> (ph <-> ps)) /\ (A e. V /\ B e. W)) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  <.cop 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568

This theorem is referenced by:  opelopabt  4699