Theorem reurab

Description: Restricted existential uniqueness of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		reurab.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2	1	bicomd	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜑 ) )
3	2	equcoms	⊢ ( 𝑦 = 𝑥 → ( 𝜓 ↔ 𝜑 ) )
4	3	elrab	⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜓 } ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
5	4	anbi1i	⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜓 } ∧ 𝜒 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜒 ) )
6		anass	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ 𝜒 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜒 ) ) )
7	5 6	bitri	⊢ ( ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜓 } ∧ 𝜒 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜒 ) ) )
8	7	eubii	⊢ ( ∃! 𝑥 ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜓 } ∧ 𝜒 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜒 ) ) )
9		df-reu	⊢ ( ∃! 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜓 } 𝜒 ↔ ∃! 𝑥 ( 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜓 } ∧ 𝜒 ) )
10		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜒 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝜑 ∧ 𝜒 ) ) )
11	8 9 10	3bitr4i	⊢ ( ∃! 𝑥 ∈ { 𝑦 ∈ 𝐴 ∣ 𝜓 } 𝜒 ↔ ∃! 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜒 ) )