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Theorem abeq2 2862
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 2867 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 4927 to inex1 4943 (look at the instance of zfauscl 4927 that occurs in the proof of inex1 4943). 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 3986 and cp 8919; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 8918. 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 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
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-5 1980 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
2 hbab1 2741 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
31, 2cleqh 2854 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}))
4 abid 2740 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
54bibi2i 326 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
65albii 1888 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥(𝑥𝐴𝜑))
73, 6bitri 264 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1622   = wceq 1624  wcel 2131  {cab 2738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748
This theorem is referenced by:  abeq1  2863  abbi2i  2868  abbi2dv  2872  clabel  2879  rabid2  3249  ru  3567  sbcabel  3650  dfss2  3724  zfrep4  4923  pwex  4989  dmopab3  5484  funimaexg  6128
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