Theorem eqabf

Description: Equality of a class variable and a class abstraction. In this version, the fact that x is a nonfree variable in A is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017) Avoid ax-13 . (Revised by Wolf Lammen, 13-May-2023)

	Ref	Expression
Hypothesis	eqabf.0	⊢ Ⅎ 𝑥 𝐴
Assertion	eqabf	⊢ ( 𝐴 = { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		eqabf.0	⊢ Ⅎ 𝑥 𝐴
2		nfab1	⊢ Ⅎ 𝑥 { 𝑥 ∣ 𝜑 }
3	1 2	cleqf	⊢ ( 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 ) )