Theorem exsb

Description: An equivalent expression for existence. One direction ( exsbim ) needs fewer axioms. (Contributed by NM, 2-Feb-2005) Avoid ax-13 . (Revised by Wolf Lammen, 16-Oct-2022)

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

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑦 𝜑
2		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 )
3		ax12v	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
4		sp	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ( 𝑥 = 𝑦 → 𝜑 ) )
5	4	com12	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → 𝜑 ) )
6	3 5	impbid	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
7	1 2 6	cbvexv1	⊢ ( ∃ 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )