Theorem dfcleq

Description: The defining characterization of class equality. It is proved, over Tarski's FOL, from the axiom of (set) extensionality ( ax-ext ) and the definition of class equality ( df-cleq ). Its forward implication is called "class extensionality". Remark: the proof uses axextb to prove also the hypothesis of df-cleq that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen , equid }. (Contributed by NM, 15-Sep-1993) (Revised by BJ, 24-Jun-2019)

	Ref	Expression
Assertion	dfcleq	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		axextb	⊢ ( 𝑦 = 𝑧 ↔ ∀ 𝑢 ( 𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧 ) )
2		axextb	⊢ ( 𝑡 = 𝑡 ↔ ∀ 𝑣 ( 𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡 ) )
3	1 2	df-cleq	⊢ ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )