Theorem abeq2

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 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 ph (that has a free variable x ) to a theorem with a class variable A , we substitute x e. A for ph throughout and simplify, where A is a new class variable not already in the wff. An example is the conversion of zfauscl to inex1 (look at the instance of zfauscl that occurs in the proof of inex1 ). Conversely, to convert a theorem with a class variable A to one with ph , we substitute { x | ph } for A throughout and simplify, where x and ph are new setvar and wff variables not already in the wff. Examples include dfsymdif2 and cp ; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 . For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. Usage of abeq2w is preferred since it requires fewer axioms. (Contributed by NM, 26-May-1993)

	Ref	Expression
Assertion	abeq2	⊢ ( 𝐴 = { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( 𝑦 ∈ 𝐴 → ∀ 𝑥 𝑦 ∈ 𝐴 )
2		hbab1	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } → ∀ 𝑥 𝑦 ∈ { 𝑥 ∣ 𝜑 } )
3	1 2	cleqh	⊢ ( 𝐴 = { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ) )
4		abid	⊢ ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 )
5	4	bibi2i	⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ) ↔ ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )
6	5	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )
7	3 6	bitri	⊢ ( 𝐴 = { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )