Theorem eubid

Description: Formula-building rule for the unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994) (Proof shortened by Wolf Lammen, 19-Feb-2023)

	Ref	Expression
Hypotheses	eubid.1	⊢ Ⅎ 𝑥 𝜑
	eubid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	eubid	⊢ ( 𝜑 → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		eubid.1	⊢ Ⅎ 𝑥 𝜑
2		eubid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1 2	alrimi	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) )
4		eubi	⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 𝜒 ) )
5	3 4	syl	⊢ ( 𝜑 → ( ∃! 𝑥 𝜓 ↔ ∃! 𝑥 𝜒 ) )