Theorem dfeumo

Description: An elementary proof showing the reverse direction of dfmoeu . Here the characterizing expression of existential uniqueness ( eu6 ) is derived from that of uniqueness ( df-mo ). (Contributed by Wolf Lammen, 3-Oct-2023)

	Ref	Expression
Assertion	dfeumo	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
2		biimpr	⊢ ( ( 𝜑 ↔ 𝑥 = 𝑦 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
3	2	aleximi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ( ∃ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝜑 ) )
4	1 3	mpi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑥 𝜑 )
5	4	exlimiv	⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) → ∃ 𝑥 𝜑 )
6	5	pm4.71ri	⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) )
7		abai	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ( ∃ 𝑥 𝜑 ∧ ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) )
8		dfmoeu	⊢ ( ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) )
9	8	anbi2i	⊢ ( ( ∃ 𝑥 𝜑 ∧ ( ∃ 𝑥 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) ) ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) )
10	6 7 9	3bitrri	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ) ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) )