Theorem 3optocl 4516

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 1145	. . . . . 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 4515	. . . . 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 4514	. . 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 1138	1 |- ( ( A e. R /\ B e. R /\ C e. R ) -> th )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 924   = wceq 1289   e. wcel 1438   <.cop 3449   X. cxp 4436

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444

This theorem is referenced by:  ecopovtrn  6389  ecopovtrng  6392