Theorem brabg 4438

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 681	. 2 |- ( ( x = A /\ y = B ) -> ( ph <-> ch ) )
4		brabg.5	. 2 |- R = { <. x , y >. | ph }
5	3, 4	brabga 4433	1 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   class class class wbr 4176   {copab 4229

This theorem is referenced by:  brab  4441  ideqg  4987  opelcnvg  5015  f1owe  6036  brrpssg  6487  bren  7080  brdomg  7081  brwdom  7495  ltprord  8867  shftfib  11846  efgrelexlema  15340  cmbr  23043  leopg  23582  cvbr  23742  mdbr  23754  dmdbr  23759  soseq  25472  sltval  25519  axcontlem5  25815  isfne  26242  isref  26253  brabg2  26311  isriscg  26494  isfrgra  28098  lcvbr  29508

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231