Theorem euimmo

Description: Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995)

	Ref	Expression
Assertion	euimmo	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		eumo	⊢ ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜓 )
2		moim	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃* 𝑥 𝜓 → ∃* 𝑥 𝜑 ) )
3	1 2	syl5	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜑 ) )