Theorem brabg 4303

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 4298	1 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   class class class wbr 4033   {copab 4093

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095

This theorem is referenced by:  brab  4307  opbrop  4742  ideqg  4817  opelcnvg  4846  bren  6806  brdomg  6807  enq0breq  7503  ltresr  7906  ltxrlt  8092  apreap  8614  apreim  8630  shftfibg  10985  shftfib  10988  2shfti  10996