Theorem reuss

Description: Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999)

	Ref	Expression
Assertion	reuss	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ 𝐵 𝜑 ) → ∃! 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝜑 → 𝜑 )
2	1	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜑 )
3		reuss2	⊢ ( ( ( 𝐴 ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜑 ) ) ∧ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) → ∃! 𝑥 ∈ 𝐴 𝜑 )
4	2 3	mpanl2	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ 𝐵 𝜑 ) ) → ∃! 𝑥 ∈ 𝐴 𝜑 )
5	4	3impb	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃! 𝑥 ∈ 𝐵 𝜑 ) → ∃! 𝑥 ∈ 𝐴 𝜑 )