Theorem eubidv

Description: Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022)

	Ref	Expression
Hypothesis	eubidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	eubidv	⊢ ( 𝜑 → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		eubidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) )
3		eubi	⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 𝜒 ) )
4	2 3	syl	⊢ ( 𝜑 → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 𝜒 ) )