Theorem elinintab

Description: Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020)

	Ref	Expression
Assertion	elinintab	⊢ ( 𝐴 ∈ ( 𝐵 ∩ ∩ { 𝑥 ∣ 𝜑 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝐴 ∈ ( 𝐵 ∩ ∩ { 𝑥 ∣ 𝜑 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ) )
2		elintabg	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )
3	2	pm5.32i	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ ∩ { 𝑥 ∣ 𝜑 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )
4	1 3	bitri	⊢ ( 𝐴 ∈ ( 𝐵 ∩ ∩ { 𝑥 ∣ 𝜑 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝜑 → 𝐴 ∈ 𝑥 ) ) )