Theorem 2ecoptocl 7838

Description: Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)

Hypotheses

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

Assertion

Ref	Expression
2ecoptocl	|- ( ( A e. S /\ B e. S ) -> ch )

Distinct variable groups:   x, y, z, w, A   z, B, w   x, C, y, z, w   x, D, y, z, w   z, S, w   x, R, y, z, w   ps, x, y   ch, z, w

Allowed substitution hints:   ph( x, y, z, w)   ps( z, w)   ch( x, y)   B( x, y)   S( x, y)

Proof of Theorem 2ecoptocl

Step	Hyp	Ref	Expression
1		2ecoptocl.1	. . 3 |- S = ( ( C X. D ) /. R )
2		2ecoptocl.3	. . . 4 |- ( [ <. z , w >. ] R = B -> ( ps <-> ch ) )
3	2	imbi2d 330	. . 3 |- ( [ <. z , w >. ] R = B -> ( ( A e. S -> ps ) <-> ( A e. S -> ch ) ) )
4		2ecoptocl.2	. . . . . 6 |- ( [ <. x , y >. ] R = A -> ( ph <-> ps ) )
5	4	imbi2d 330	. . . . 5 |- ( [ <. x , y >. ] R = A -> ( ( ( z e. C /\ w e. D ) -> ph ) <-> ( ( z e. C /\ w e. D ) -> ps ) ) )
6		2ecoptocl.4	. . . . . 6 |- ( ( ( x e. C /\ y e. D ) /\ ( z e. C /\ w e. D ) ) -> ph )
7	6	ex 450	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( ( z e. C /\ w e. D ) -> ph ) )
8	1, 5, 7	ecoptocl 7837	. . . 4 |- ( A e. S -> ( ( z e. C /\ w e. D ) -> ps ) )
9	8	com12 32	. . 3 |- ( ( z e. C /\ w e. D ) -> ( A e. S -> ps ) )
10	1, 3, 9	ecoptocl 7837	. 2 |- ( B e. S -> ( A e. S -> ch ) )
11	10	impcom 446	1 |- ( ( A e. S /\ B e. S ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   X. cxp 5112   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  3ecoptocl  7839  ecovcom  7854  addclsr  9904  mulclsr  9905  ltsosr  9915  mulgt0sr  9926