Theorem 3optocl 4474

Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

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

Allowed substitution hints:   ph( x, y, z, w, v, u)   ps( z, w, v, u)   ch( x, y, v, u)   th( x, y, z, w)   B( x, y)   C( x, y, z, w)   R( x, y)

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 228	. . . 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 228	. . . . . 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 228	. . . . . 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 1141	. . . . . 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 4473	. . . . 5 |- ( ( A e. R /\ B e. R ) -> ( ( v e. D /\ u e. F ) -> ch ) )
11	10	com12 30	. . . 4 |- ( ( v e. D /\ u e. F ) -> ( ( A e. R /\ B e. R ) -> ch ) )
12	1, 3, 11	optocl 4472	. . 3 |- ( C e. R -> ( ( A e. R /\ B e. R ) -> th ) )
13	12	impcom 123	. 2 |- ( ( ( A e. R /\ B e. R ) /\ C e. R ) -> th )
14	13	3impa 1134	1 |- ( ( A e. R /\ B e. R /\ C e. R ) -> th )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   <.cop 3425   X. cxp 4399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4407

This theorem is referenced by:  ecopovtrn  6319  ecopovtrng  6322