Theorem eubii

Description: Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994) (Revised by Mario Carneiro, 6-Oct-2016)

	Ref	Expression
Hypothesis	eubii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	eubii	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		eubii.1	⊢ ( 𝜑 ↔ 𝜓 )
2		eubi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∃! 𝑥 𝜑 ↔ ∃! 𝑥 𝜓 ) )
3	2 1	mpg	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑥 𝜓 )