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Theorem abeq2 2128
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 2133 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 φ (that has a free variable x) to a theorem with a class variable A, we substitute x A for φ throughout and simplify, where A is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable A to one with φ, we substitute {xφ} for A throughout and simplify, where x and φ are new set and wff variables not already in the wff. 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 = {xφ} ↔ x(x Aφ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem abeq2
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ax-17 1401 . . 3 (y Ax y A)
2 hbab1 2011 . . 3 (y {xφ} → x y {xφ})
31, 2cleqh 2119 . 2 (A = {xφ} ↔ x(x Ax {xφ}))
4 abid 2010 . . . 4 (x {xφ} ↔ φ)
54bibi2i 216 . . 3 ((x Ax {xφ}) ↔ (x Aφ))
65albii 1338 . 2 (x(x Ax {xφ}) ↔ x(x Aφ))
73, 6bitri 173 1 (A = {xφ} ↔ x(x Aφ))
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1314   = wceq 1373   wcel 1375  {cab 2008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-11 1379  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018
This theorem is referenced by:  abeq1  2129  abbi2i  2134  abbi2dv  2138  clabel  2145  sbabel  2185  rabid2  2462  ru  2738  sbcabel  2815  dfss2  2912  pwex  3884  dmopab3  4441
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