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Theorem abeq2 2509
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 2514 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 4292 to inex1 4304 (look at the instance of zfauscl 4292 that occurs in the proof of inex1 4304). 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 7771, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 7770. 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
Allowed substitution hint:    ph( x)

Proof of Theorem abeq2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax-17 1623 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
2 hbab1 2393 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2cleqh 2501 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
4 abid 2392 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
54bibi2i 305 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
65albii 1572 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
73, 6bitri 241 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1721   {cab 2390
This theorem is referenced by:  abeq1  2510  abbi2i  2515  abbi2dv  2519  clabel  2525  sbabel  2566  rabid2  2845  ru  3120  sbcabel  3198  dfss2  3297  zfrep4  4288  pwex  4342  dmopab3  5041  funimaexg  5489
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400
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