Theorem exsb 2456

Description: An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exsb

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑦𝜑
2		nfa1 2015	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑)
3		ax12v 2035	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		sp 2041	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
5	4	com12 32	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
6	3, 5	impbid 201	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	1, 2, 6	cbvex 2260	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)