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Theorem oprabbi 32108
Description: Equality deduction for class abstraction of nested ordered pairs. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Assertion
Ref Expression
oprabbi  |-  ( A. x A. y A. z
( ph  <->  ps )  ->  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps } )

Proof of Theorem oprabbi
StepHypRef Expression
1 eqoprab2b 6363 . 2  |-  ( {
<. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. <. x ,  y
>. ,  z >.  |  ps }  <->  A. x A. y A. z (
ph 
<->  ps ) )
21biimpri 209 1  |-  ( A. x A. y A. z
( ph  <->  ps )  ->  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { <. <. x ,  y >. ,  z
>.  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437   {coprab 6306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-oprab 6309
This theorem is referenced by:  mpt2bi123f  32109
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