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Theorem brabsb 4292
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 4040 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabsb.1 . . 3  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2360 . 2  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
4 opelopabsb 4291 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
51, 3, 43bitri 262 1  |-  ( A R B  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1632    e. wcel 1696   [.wsbc 3004   <.cop 3656   class class class wbr 4039   {copab 4092
This theorem is referenced by:  eqerlem  6708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094
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