Theorem euim

Description: Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005) (Proof shortened by Andrew Salmon, 14-Jun-2011) (Proof shortened by Wolf Lammen, 1-Oct-2023)

	Ref	Expression
Assertion	euim	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) → ( ∃! 𝑥 𝜓 → ∃! 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		euimmo	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 𝜓 → ∃* 𝑥 𝜑 ) )
2		exmoeub	⊢ ( ∃ 𝑥 𝜑 → ( ∃* 𝑥 𝜑 ↔ ∃! 𝑥 𝜑 ) )
3	2	biimpd	⊢ ( ∃ 𝑥 𝜑 → ( ∃* 𝑥 𝜑 → ∃! 𝑥 𝜑 ) )
4	1 3	sylan9r	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝜑 → 𝜓 ) ) → ( ∃! 𝑥 𝜓 → ∃! 𝑥 𝜑 ) )