Theorem 2optocl 4852

Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Hypotheses

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

Assertion

Ref	Expression
2optocl	|- ( ( A e. R /\ B e. R ) -> ch )

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

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

Proof of Theorem 2optocl

Step	Hyp	Ref	Expression
1		2optocl.1	. . 3 |- R = ( C X. D )
2		2optocl.3	. . . 4 |- ( <. z , w >. = B -> ( ps <-> ch ) )
3	2	imbi2d 230	. . 3 |- ( <. z , w >. = B -> ( ( A e. R -> ps ) <-> ( A e. R -> ch ) ) )
4		2optocl.2	. . . . . 6 |- ( <. x , y >. = A -> ( ph <-> ps ) )
5	4	imbi2d 230	. . . . 5 |- ( <. x , y >. = A -> ( ( ( z e. C /\ w e. D ) -> ph ) <-> ( ( z e. C /\ w e. D ) -> ps ) ) )
6		2optocl.4	. . . . . 6 |- ( ( ( x e. C /\ y e. D ) /\ ( z e. C /\ w e. D ) ) -> ph )
7	6	ex 115	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( ( z e. C /\ w e. D ) -> ph ) )
8	1, 5, 7	optocl 4851	. . . 4 |- ( A e. R -> ( ( z e. C /\ w e. D ) -> ps ) )
9	8	com12 30	. . 3 |- ( ( z e. C /\ w e. D ) -> ( A e. R -> ps ) )
10	1, 3, 9	optocl 4851	. 2 |- ( B e. R -> ( A e. R -> ch ) )
11	10	impcom 125	1 |- ( ( A e. R /\ B e. R ) -> ch )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3712   X. cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-xp 4780

This theorem is used by:  3optocl  4853  ecopovsym  6905  ecopovsymg  6908  th3qlem2  6912  axaddcom  8237