Theorem brabg2 28750

Description: Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
brabg2.1	|- ( x = A -> ( ph <-> ps ) )
brabg2.2	|- ( y = B -> ( ps <-> ch ) )
brabg2.3	|- R = { <. x , y >. | ph }
brabg2.4	|- ( ch -> A e. C )

Assertion

Ref	Expression
brabg2	|- ( B e. D -> ( A R B <-> ch ) )

Distinct variable groups:   x, A, y   x, B, y   ch, x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   C( x, y)   D( x, y)   R( x, y)

Proof of Theorem brabg2

Step	Hyp	Ref	Expression
1		brabg2.3	. . . . 5 |- R = { <. x , y >. | ph }
2	1	relopabi 5066	. . . 4 |- Rel R
3	2	brrelexi 4980	. . 3 |- ( A R B -> A e. _V )
4		brabg2.1	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
5		brabg2.2	. . . . . . 7 |- ( y = B -> ( ps <-> ch ) )
6	4, 5, 1	brabg 4709	. . . . . 6 |- ( ( A e. _V /\ B e. D ) -> ( A R B <-> ch ) )
7	6	biimpd 207	. . . . 5 |- ( ( A e. _V /\ B e. D ) -> ( A R B -> ch ) )
8	7	ex 434	. . . 4 |- ( A e. _V -> ( B e. D -> ( A R B -> ch ) ) )
9	8	com3l 81	. . 3 |- ( B e. D -> ( A R B -> ( A e. _V -> ch ) ) )
10	3, 9	mpdi 42	. 2 |- ( B e. D -> ( A R B -> ch ) )
11		brabg2.4	. . 3 |- ( ch -> A e. C )
12	4, 5, 1	brabg 4709	. . . . 5 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )
13	12	exbiri 622	. . . 4 |- ( A e. C -> ( B e. D -> ( ch -> A R B ) ) )
14	13	com3l 81	. . 3 |- ( B e. D -> ( ch -> ( A e. C -> A R B ) ) )
15	11, 14	mpdi 42	. 2 |- ( B e. D -> ( ch -> A R B ) )
16	10, 15	impbid 191	1 |- ( B e. D -> ( A R B <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3071   class class class wbr 4393   {copab 4450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948

This theorem is referenced by: (None)