Theorem rexab

Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014) (Revised by Mario Carneiro, 3-Sep-2015)

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

Proof

Step	Hyp	Ref	Expression
1		ralab.1	⊢ ( 𝑦 = 𝑥 → ( 𝜑 ↔ 𝜓 ) )
2		df-rex	⊢ ( ∃ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∃ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ∧ 𝜒 ) )
3		vex	⊢ 𝑥 ∈ V
4	3 1	elab	⊢ ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ↔ 𝜓 )
5	4	anbi1i	⊢ ( ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ∧ 𝜒 ) ↔ ( 𝜓 ∧ 𝜒 ) )
6	5	exbii	⊢ ( ∃ 𝑥 ( 𝑥 ∈ { 𝑦 ∣ 𝜑 } ∧ 𝜒 ) ↔ ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )
7	2 6	bitri	⊢ ( ∃ 𝑥 ∈ { 𝑦 ∣ 𝜑 } 𝜒 ↔ ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )