Theorem 3ecoptocl 6836

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 1233	. . . . . 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 6834	. . . . 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 6835	. . 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 1228	1 |- ( ( A e. S /\ B e. S /\ C e. S ) -> th )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   <.cop 3676   X. cxp 4729   [cec 6743   /.cqs 6744

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-ec 6747  df-qs 6751

This theorem is referenced by:  ecovass  6856  ecoviass  6857  ecovdi  6858  ecovidi  6859  ltsonq  7678  ltanqg  7680  ltmnqg  7681  lttrsr  8042  ltsosr  8044  ltasrg  8050  mulextsr1  8061