Theorem rabbi2dva

Description: Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014)

	Ref	Expression
Hypothesis	rabbi2dva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ 𝐵 ↔ 𝜓 ) )
Assertion	rabbi2dva	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∈ 𝐴 ∣ 𝜓 } )

Proof

Step	Hyp	Ref	Expression
1		rabbi2dva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ 𝐵 ↔ 𝜓 ) )
2		dfin5	⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵 }
3	1	rabbidva	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵 } = { 𝑥 ∈ 𝐴 ∣ 𝜓 } )
4	2 3	eqtrid	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = { 𝑥 ∈ 𝐴 ∣ 𝜓 } )