Theorem reuanid

Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018) (Proof shortened by Wolf Lammen, 12-Jan-2025)

	Ref	Expression
Assertion	reuanid	⊢ ( ∃! 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃! 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ibar	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2	1	bicomd	⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ 𝜑 ) )
3	2	reubiia	⊢ ( ∃! 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃! 𝑥 ∈ 𝐴 𝜑 )