Theorem eliminable3b

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment). (Contributed by BJ, 19-Oct-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	eliminable3b	⊢ ( { 𝑥 ∣ 𝜑 } ∈ { 𝑦 ∣ 𝜓 } ↔ ∃ 𝑧 ( 𝑧 = { 𝑥 ∣ 𝜑 } ∧ 𝑧 ∈ { 𝑦 ∣ 𝜓 } ) )

Proof

Step	Hyp	Ref	Expression
1		dfclel	⊢ ( { 𝑥 ∣ 𝜑 } ∈ { 𝑦 ∣ 𝜓 } ↔ ∃ 𝑧 ( 𝑧 = { 𝑥 ∣ 𝜑 } ∧ 𝑧 ∈ { 𝑦 ∣ 𝜓 } ) )