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Theorem abeq2 2578
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 2585 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 4526 to inex1 4544 (look at the instance of zfauscl 4526 that occurs in the proof of inex1 4544). 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 setvar and wff variables not already in the wff. An example is cp 8213, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 8212. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 26-May-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-5 1671 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
2 hbab1 2442 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
31, 2cleqh 2569 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
4 abid 2441 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
54bibi2i 313 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
65albii 1611 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
73, 6bitri 249 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368    = wceq 1370    e. wcel 1758   {cab 2439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449
This theorem is referenced by:  abeq1  2579  abbi2i  2587  abbi2dv  2591  clabel  2600  sbabel  2647  rabid2  3004  ru  3293  sbcabel  3383  dfss2  3456  zfrep4  4522  pwex  4586  dmopab3  5163  funimaexg  5606
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