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Theorem oprabbii 5919
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 2296 . 2  |-  w  =  w
2 oprabbii.1 . . . 4  |-  ( ph  <->  ps )
32a1i 10 . . 3  |-  ( w  =  w  ->  ( ph 
<->  ps ) )
43oprabbidv 5918 . 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 176    = wceq 1632   {coprab 5875
This theorem is referenced by:  oprab4  5933  mpt2v  5953  dfxp3  6195  tposmpt2  6287  ovec  6784  addcnsr  8773  mulcnsr  8774  prismorcsetlemc  26020
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-oprab 5878
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