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Theorem cbvoprab12v 6271
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)
Hypothesis
Ref Expression
cbvoprab12v.1  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbvoprab12v  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  v >. ,  z
>.  |  ps }
Distinct variable groups:    x, y,
z, w, v    ph, w, v    ps, x, y
Allowed substitution hints:    ph( x, y, z)    ps( z, w, v)

Proof of Theorem cbvoprab12v
StepHypRef Expression
1 nfv 1715 . 2  |-  F/ w ph
2 nfv 1715 . 2  |-  F/ v
ph
3 nfv 1715 . 2  |-  F/ x ps
4 nfv 1715 . 2  |-  F/ y ps
5 cbvoprab12v.1 . 2  |-  ( ( x  =  w  /\  y  =  v )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbvoprab12 6270 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. w ,  v >. ,  z
>.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   {coprab 6197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-opab 4426  df-oprab 6200
This theorem is referenced by:  cpnnen  13964
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