Theorem 3optocl 2471

Description: Implicit substitution of classes for ordered pairs.

Hypotheses

Ref	Expression
3optocl.1	|- R = (D X. F)
3optocl.2	|- (<.x, y>. = A -> (ph <-> ps))
3optocl.3	|- (<.z, w>. = B -> (ps <-> ch))
3optocl.4	|- (<.v, u>. = C -> (ch <-> th))
3optocl.5	|- (((x e. D /\ y e. F) /\ (z e. D /\ w e. F) /\ (v e. D /\ u e. F)) -> ph)

Assertion

Ref	Expression
3optocl	|- ((A e. R /\ B e. R /\ C e. R) -> th)

Distinct variable group(s):   x,y,z,w,v,u,A   z,B,w,v,u   v,C,u   x,D,y,z,w,v,u   x,F,y,z,w,v,u   z,R,w,v,u   ps,x,y   ch,z,w   th,v,u

Proof of Theorem 3optocl

Step	Hyp	Ref	Expression
1		3optocl.1	. . . . 5 |- R = (D X. F)
2		3optocl.4	. . . . . 6 |- (<.v, u>. = C -> (ch <-> th))
3	2	imbi2d 464	. . . . 5 |- (<.v, u>. = C -> (((A e. R /\ B e. R) -> ch) <-> ((A e. R /\ B e. R) -> th)))
4		3optocl.2	. . . . . . . 8 |- (<.x, y>. = A -> (ph <-> ps))
5	4	imbi2d 464	. . . . . . 7 |- (<.x, y>. = A -> (((v e. D /\ u e. F) -> ph) <-> ((v e. D /\ u e. F) -> ps)))
6		3optocl.3	. . . . . . . 8 |- (<.z, w>. = B -> (ps <-> ch))
7	6	imbi2d 464	. . . . . . 7 |- (<.z, w>. = B -> (((v e. D /\ u e. F) -> ps) <-> ((v e. D /\ u e. F) -> ch)))
8		3optocl.5	. . . . . . . . 9 |- (((x e. D /\ y e. F) /\ (z e. D /\ w e. F) /\ (v e. D /\ u e. F)) -> ph)
9	8	3expa 612	. . . . . . . 8 |- ((((x e. D /\ y e. F) /\ (z e. D /\ w e. F)) /\ (v e. D /\ u e. F)) -> ph)
10	9	exp 291	. . . . . . 7 |- (((x e. D /\ y e. F) /\ (z e. D /\ w e. F)) -> ((v e. D /\ u e. F) -> ph))
11	1, 5, 7, 10	2optocl 2470	. . . . . 6 |- ((A e. R /\ B e. R) -> ((v e. D /\ u e. F) -> ch))
12	11	com12 13	. . . . 5 |- ((v e. D /\ u e. F) -> ((A e. R /\ B e. R) -> ch))
13	1, 3, 12	optocl 2469	. . . 4 |- (C e. R -> ((A e. R /\ B e. R) -> th))
14	13	com12 13	. . 3 |- ((A e. R /\ B e. R) -> (C e. R -> th))
15	14	imp 277	. 2 |- (((A e. R /\ B e. R) /\ C e. R) -> th)
16	15	3impa 609	1 |- ((A e. R /\ B e. R /\ C e. R) -> th)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092  <.cop 1810   X. cxp 2408

This theorem is referenced by:  ecopoprtrn 3247

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-3an 583  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  df-opab 2098  df-xp 2424

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