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Theorem eqabb 2881
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 abbib 2811 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 𝑥𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. An example is the conversion of zfauscl 5298 to inex1 5317 (look at the instance of zfauscl 5298 that occurs in the proof of inex1 5317). 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 setvar and wff variables not already in the wff. Examples include dfsymdif2 4261 and cp 9931; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 9930. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.

Usage of eqabbw 2815 is preferred since it requires fewer axioms. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2025.)

Assertion
Ref Expression
eqabb (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eqabb
StepHypRef Expression
1 abid1 2878 . . 3 𝐴 = {𝑥𝑥𝐴}
21eqeq1i 2742 . 2 (𝐴 = {𝑥𝜑} ↔ {𝑥𝑥𝐴} = {𝑥𝜑})
3 abbib 2811 . 2 ({𝑥𝑥𝐴} = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3bitri 275 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538   = wceq 1540  wcel 2108  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816
This theorem is referenced by:  eqabcb  2883  clabel  2888  rabid2OLD  3471  ruOLD  3787  sbcabel  3878  zfrep4  5293  dmopab3  5930  rnopab3  5967  funimaexgOLD  6654  fineqvrep  35109  bj-abex  37031  sticksstones1  42147  sticksstones2  42148
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