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Theorem abeq2 2389
Description: Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2394 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable.

Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable  ph (that has a free variable  x) to a theorem with a class variable  A, we substitute  x  e.  A for  ph throughout and simplify, where  A is a new class variable not already in the wff. An example is the conversion of zfauscl 4144 to inex1 4156 (look at the instance of zfauscl 4144 that occurs in the proof of inex1 4156). Conversely, to convert a theorem with a class variable  A to one with 
ph, we substitute  { x  | 
ph } for  A throughout and simplify, where  x and  ph are new set and wff variables not already in the wff. An example is cp 7556, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 7555. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
abeq2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Distinct variable group:    x, A
Dummy variable  y is distinct from all other variables.
Allowed substitution hint:    ph( x)

Proof of Theorem abeq2
StepHypRef Expression
1 ax-17 1604 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
2 hbab1 2273 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2cleqh 2381 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
4 abid 2272 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
54bibi2i 306 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
65albii 1554 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
73, 6bitri 242 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1528    = wceq 1624    e. wcel 1685   {cab 2270
This theorem is referenced by:  abeq1  2390  abbi2i  2395  abbi2dv  2399  clabel  2405  sbabel  2446  rabid2  2718  ru  2991  sbcabel  3069  dfss2  3170  zfrep4  4140  pwex  4192  dmopab3  4890  funimaexg  5294
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280
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