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Metamath Proof Explorer

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 ∈ 𝐴 → 𝜑 )
	Assertion	rabeqc	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = 𝐴

Proof

Step	Hyp	Ref	Expression
1		rabeqc.1	⊢ ( 𝑥 ∈ 𝐴 → 𝜑 )
2	1	adantl	⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐴 ) → 𝜑 )
3	2	rabeqcda	⊢ ( ⊤ → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = 𝐴 )
4	3	mptru	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = 𝐴