MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprabbii Structured version   Unicode version

Theorem oprabbii 6121
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 2435 . 2  |-  w  =  w
2 oprabbii.1 . . . 4  |-  ( ph  <->  ps )
32a1i 11 . . 3  |-  ( w  =  w  ->  ( ph 
<->  ps ) )
43oprabbidv 6120 . 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 1652   {coprab 6074
This theorem is referenced by:  oprab4  6135  mpt2v  6155  dfxp3  6398  tposmpt2  6508  ovec  7006  addcnsr  9000  mulcnsr  9001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-oprab 6077
  Copyright terms: Public domain W3C validator