Theorem rabeqc

Description: A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022) (Proof shortened by SN, 15-Jan-2025)

	Ref	Expression
Hypothesis	rabeqc.1	$$⊢x\in A\to \phi$$
Assertion	rabeqc	$$⊢\left\{x\in A|\phi \right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		rabeqc.1	$$⊢x\in A\to \phi$$
2	1	adantl	$$⊢\left(\top \wedge x\in A\right)\to \phi$$
3	2	rabeqcda	$$⊢\top \to \left\{x\in A|\phi \right\}=A$$
4	3	mptru	$$⊢\left\{x\in A|\phi \right\}=A$$