Theorem brabg 4256

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 683	. 2 |- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
4		brabg.5	. 2 |- R = { <. x , y >. | ph }
5	3, 4	brabga 4251	1 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )

Syntax hints:   -> wi 6   <-> wb 178   /\ wa 360   = wceq 1619   e. wcel 1621   class class class wbr 3997   {copab 4050

This theorem is referenced by:  brab  4259  ideqg  4823  opelcnvg  4849  f1owe  5784  brrpssg  6213  bren  6839  brdomg  6840  brwdom  7249  ltprord  8622  shftfib  11533  efgrelexlema  15021  cmbr  22124  leopg  22663  cvbr  22823  mdbr  22835  dmdbr  22840  soseq  23624  sltval  23671  axcontlem5  23972  isfne  25636  isref  25647  brabg2  25734  isriscg  25983  lcvbr  28461

This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052