Theorem 3ecoptocl 6283

Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)

Hypotheses

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

Assertion

Ref	Expression
3ecoptocl	|- ( ( A e. S /\ B e. S /\ C e. S ) -> 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   z, S, w, v, u   x, R, y, z, 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)   S( x, y)

Proof of Theorem 3ecoptocl

Step	Hyp	Ref	Expression
1		3ecoptocl.1	. . . 4 |- S = ( ( D X. D ) /. R )
2		3ecoptocl.3	. . . . 5 |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) )
3	2	imbi2d 228	. . . 4 |- ( [ <. z , w >. ] R = B -> ( ( A e. S -> ps ) <-> ( A e. S -> ch ) ) )
4		3ecoptocl.4	. . . . 5 |- ( [ <. v , u >. ] R = C -> ( ch <-> th ) )
5	4	imbi2d 228	. . . 4 |- ( [ <. v , u >. ] R = C -> ( ( A e. S -> ch ) <-> ( A e. S -> th ) ) )
6		3ecoptocl.2	. . . . . . 7 |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) )
7	6	imbi2d 228	. . . . . 6 |- ( [ <. x , y >. ] R = A -> ( ( ( ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ph ) <-> ( ( ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ps ) ) )
8		3ecoptocl.5	. . . . . . 7 |- ( ( ( x e. D /\ y e. D ) /\ ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ph )
9	8	3expib 1142	. . . . . 6 |- ( ( x e. D /\ y e. D ) -> ( ( ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ph ) )
10	1, 7, 9	ecoptocl 6281	. . . . 5 |- ( A e. S -> ( ( ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ps ) )
11	10	com12 30	. . . 4 |- ( ( ( z e. D /\ w e. D ) /\ ( v e. D /\ u e. D ) ) -> ( A e. S -> ps ) )
12	1, 3, 5, 11	2ecoptocl 6282	. . 3 |- ( ( B e. S /\ C e. S ) -> ( A e. S -> th ) )
13	12	com12 30	. 2 |- ( A e. S -> ( ( B e. S /\ C e. S ) -> th ) )
14	13	3impib 1137	1 |- ( ( A e. S /\ B e. S /\ C e. S ) -> th )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   <.cop 3420   X. cxp 4390   [cec 6192   /.cqs 6193

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 3917  ax-pow 3969  ax-pr 3993

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 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-xp 4398  df-cnv 4400  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-ec 6196  df-qs 6200

This theorem is referenced by:  ecovass  6303  ecoviass  6304  ecovdi  6305  ecovidi  6306  ltsonq  6686  ltanqg  6688  ltmnqg  6689  lttrsr  7037  ltsosr  7039  ltasrg  7045  mulextsr1  7055