Theorem 2optocl 4688

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 229	. . 3 |- ( <. z , w >. = B -> ( ( A e. R -> ps ) <-> ( A e. R -> ch ) ) )
4		2optocl.2	. . . . . 6 |- ( <. x , y >. = A -> ( ph <-> ps ) )
5	4	imbi2d 229	. . . . 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 114	. . . . 5 |- ( ( x e. C /\ y e. D ) -> ( ( z e. C /\ w e. D ) -> ph ) )
8	1, 5, 7	optocl 4687	. . . 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 4687	. 2 |- ( B e. R -> ( A e. R -> ch ) )
11	10	impcom 124	1 |- ( ( A e. R /\ B e. R ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   <.cop 3586   X. cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617

This theorem is referenced by:  3optocl  4689  ecopovsym  6609  ecopovsymg  6612  th3qlem2  6616  axaddcom  7832