Theorem rexrab

Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011) (Revised by Mario Carneiro, 3-Sep-2015)

	Ref	Expression
Hypothesis	ralab.1	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rexrab	⊢ ( ∃ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜒 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralab.1	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
2	1	elrab	⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
3	2	anbi1i	⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } ∧ 𝜒 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ∧ 𝜒 ) )
4		anass	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜓 ∧ 𝜒 ) ) )
5	3 4	bitri	⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } ∧ 𝜒 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜓 ∧ 𝜒 ) ) )
6	5	rexbii2	⊢ ( ∃ 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜑 } 𝜒 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝜓 ∧ 𝜒 ) )