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Theorem brabsb 4747
Description: The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
Hypothesis
Ref Expression
brabsb.1  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabsb  |-  ( A R B  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)    R( x, y)

Proof of Theorem brabsb
StepHypRef Expression
1 df-br 4440 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabsb.1 . . 3  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2532 . 2  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
4 opelopabsb 4746 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
51, 3, 43bitri 271 1  |-  ( A R B  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1398    e. wcel 1823   [.wsbc 3324   <.cop 4022   class class class wbr 4439   {copab 4496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498
This theorem is referenced by:  eqerlem  7335
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