Theorem abeq2 2221

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 2226 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 {𝑥 ∣ 𝜑} 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 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abeq2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1487	. . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
2		hbab1 2102	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
3	1, 2	cleqh 2212	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
4		abid 2101	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5	4	bibi2i 226	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
6	5	albii 1427	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
7	3, 6	bitri 183	1 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Syntax hints:   ↔ wb 104  ∀wal 1310   = wceq 1312   ∈ wcel 1461  {cab 2099

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-11 1465  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109

This theorem is referenced by:  abeq1  2222  abbi2i  2227  abbi2dv  2231  clabel  2238  sbabel  2279  rabid2  2579  ru  2875  sbcabel  2956  dfss2  3050  vpwex  4061  dmopab3  4710