Theorem mo4f

Description: At-most-one quantifier expressed using implicit substitution. Note that the disjoint variable condition on y , ph can be replaced by the nonfreeness hypothesis |- F/ y ph with essentially the same proof. (Contributed by NM, 10-Apr-2004) Remove dependency on ax-13 . (Revised by Wolf Lammen, 19-Jan-2023)

	Ref	Expression
Hypotheses	mo4f.1	⊢ Ⅎ 𝑥 𝜓
	mo4f.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	mo4f	⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		mo4f.1	⊢ Ⅎ 𝑥 𝜓
2		mo4f.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		nfv	⊢ Ⅎ 𝑦 𝜑
4	3	mo3	⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
5	1 2	sbiev	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
6	5	anbi2i	⊢ ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) )
7	6	imbi1i	⊢ ( ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝑦 ) )
8	7	2albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝑦 ) )
9	4 8	bitri	⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝑦 ) )