Theorem dfmo

Description: Rederive df-mo from the old definition moeu . (Contributed by Wolf Lammen, 27-May-2019) (Proof modification is discouraged.) Use df-mo instead. (New usage is discouraged.)

	Ref	Expression
Assertion	dfmo	⊢ ( ∃* 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		moeu	⊢ ( ∃* 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) )
2		eu6	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) )
3	2	imbi2i	⊢ ( ( ∃ 𝑥 𝜑 → ∃! 𝑥 𝜑 ) ↔ ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) )
4		dfmoeu	⊢ ( ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) )
5	1 3 4	3bitri	⊢ ( ∃* 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) )