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Theorem oprabbii 6069
Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
oprabbii.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
oprabbii  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps }
Distinct variable groups:    x, z    y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem oprabbii
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 eqid 2388 . 2  |-  w  =  w
2 oprabbii.1 . . . 4  |-  ( ph  <->  ps )
32a1i 11 . . 3  |-  ( w  =  w  ->  ( ph 
<->  ps ) )
43oprabbidv 6068 . 2  |-  ( w  =  w  ->  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps } )
51, 4ax-mp 8 1  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps }
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   {coprab 6022
This theorem is referenced by:  oprab4  6083  mpt2v  6103  dfxp3  6346  tposmpt2  6453  ovec  6951  addcnsr  8944  mulcnsr  8945
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-oprab 6025
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