Theorem 3optocl 3243

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 groups:   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	. . . 4 |- R = (D X. F)
2		3optocl.4	. . . . 5 |- (<.v, u>. = C -> (ch <-> th))
3	2	imbi2d 614	. . . 4 |- (<.v, u>. = C -> (((A e. R /\ B e. R) -> ch) <-> ((A e. R /\ B e. R) -> th)))
4		3optocl.2	. . . . . . 7 |- (<.x, y>. = A -> (ph <-> ps))
5	4	imbi2d 614	. . . . . 6 |- (<.x, y>. = A -> (((v e. D /\ u e. F) -> ph) <-> ((v e. D /\ u e. F) -> ps)))
6		3optocl.3	. . . . . . 7 |- (<.z, w>. = B -> (ps <-> ch))
7	6	imbi2d 614	. . . . . 6 |- (<.z, w>. = B -> (((v e. D /\ u e. F) -> ps) <-> ((v e. D /\ u e. F) -> ch)))
8		3optocl.5	. . . . . . 7 |- (((x e. D /\ y e. F) /\ (z e. D /\ w e. F) /\ (v e. D /\ u e. F)) -> ph)
9	8	3expia 837	. . . . . 6 |- (((x e. D /\ y e. F) /\ (z e. D /\ w e. F)) -> ((v e. D /\ u e. F) -> ph))
10	1, 5, 7, 9	2optocl 3242	. . . . 5 |- ((A e. R /\ B e. R) -> ((v e. D /\ u e. F) -> ch))
11	10	com12 11	. . . 4 |- ((v e. D /\ u e. F) -> ((A e. R /\ B e. R) -> ch))
12	1, 3, 11	optocl 3241	. . 3 |- (C e. R -> ((A e. R /\ B e. R) -> th))
13	12	impcom 351	. 2 |- (((A e. R /\ B e. R) /\ C e. R) -> th)
14	13	3impa 830	1 |- ((A e. R /\ B e. R /\ C e. R) -> th)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  <.cop 2415   X. cxp 3174

This theorem is referenced by:  ecopoprtrn 4317

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-3an 779  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  df-opab 2672  df-xp 3190

Copyright terms: Public domain