Theorem brabg 4387

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 4382	1 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109   {copab 4170

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172

This theorem is referenced by:  brab  4391  opbrop  4829  ideqg  4906  opelcnvg  4935  breng  6982  bren  6983  brdom2g  6984  brdomg  6985  enq0breq  7751  ltresr  8154  ltxrlt  8339  apreap  8861  apreim  8877  shftfibg  11505  shftfib  11508  2shfti  11516