Theorem 3ecoptocl 6711

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 1209	. . . . . 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 6709	. . . . 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 6710	. . 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 1204	1 |- ( ( A e. S /\ B e. S /\ C e. S ) -> th )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   <.cop 3636   X. cxp 4673   [cec 6618   /.cqs 6619

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-ec 6622  df-qs 6626

This theorem is referenced by:  ecovass  6731  ecoviass  6732  ecovdi  6733  ecovidi  6734  ltsonq  7511  ltanqg  7513  ltmnqg  7514  lttrsr  7875  ltsosr  7877  ltasrg  7883  mulextsr1  7894