Theorem brabg 4270

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopabg.1	|- ( x = A -> ( ph <-> ps ) )
opelopabg.2	|- ( y = B -> ( ps <-> ch ) )
brabg.5	|- R = { <. x , y >. | ph }

Assertion

Ref	Expression
brabg	|- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )

Distinct variable groups:   x, y, A   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 brabg

Step	Hyp	Ref	Expression
1		opelopabg.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2		opelopabg.2	. . 3 |- ( y = B -> ( ps <-> ch ) )
3	1, 2	sylan9bb 462	. 2 |- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
4		brabg.5	. 2 |- R = { <. x , y >. | ph }
5	3, 4	brabga 4265	1 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   class class class wbr 4004   {copab 4064

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066

This theorem is referenced by:  brab  4273  opbrop  4706  ideqg  4779  opelcnvg  4808  bren  6747  brdomg  6748  enq0breq  7435  ltresr  7838  ltxrlt  8023  apreap  8544  apreim  8560  shftfibg  10829  shftfib  10832  2shfti  10840