Theorem 3ecoptocl 6792

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 230	. . . 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 230	. . . 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 230	. . . . . 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 1232	. . . . . 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 6790	. . . . 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 6791	. . 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 1227	1 |- ( ( A e. S /\ B e. S /\ C e. S ) -> th )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   <.cop 3672   X. cxp 4723   [cec 6699   /.cqs 6700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-ec 6703  df-qs 6707

This theorem is referenced by:  ecovass  6812  ecoviass  6813  ecovdi  6814  ecovidi  6815  ltsonq  7617  ltanqg  7619  ltmnqg  7620  lttrsr  7981  ltsosr  7983  ltasrg  7989  mulextsr1  8000