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Theorem abeq2 2581
 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 2588 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 (that has a free variable ) to a theorem with a class variable , we substitute for throughout and simplify, where is a new class variable not already in the wff. An example is the conversion of zfauscl 4575 to inex1 4593 (look at the instance of zfauscl 4575 that occurs in the proof of inex1 4593). Conversely, to convert a theorem with a class variable to one with , we substitute for throughout and simplify, where and are new setvar and wff variables not already in the wff. An example is cp 8330, which derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 8329. 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
Distinct variable group:   ,

Proof of Theorem abeq2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-5 1704 . . 3
2 hbab1 2445 . . 3
31, 2cleqh 2572 . 2
4 abid 2444 . . . 4
54bibi2i 313 . . 3
65albii 1640 . 2
73, 6bitri 249 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  A.wal 1393  =wceq 1395  e.wcel 1818  {cab 2442 This theorem is referenced by:  abeq1  2582  abbi2i  2590  abbi2dv  2594  clabel  2603  sbabelOLD  2651  rabid2  3035  ru  3326  sbcabel  3416  dfss2  3492  zfrep4  4571  pwex  4635  dmopab3  5220  funimaexg  5670 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452
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