Theorem reuanid

Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018)

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

Proof

Step	Hyp	Ref	Expression
1		anabs5	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
2	1	eubii	⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
3		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
4		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
5	2 3 4	3bitr4i	⊢ ( ∃! 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃! 𝑥 ∈ 𝐴 𝜑 )