Theorem exsb 2130

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 1619	. 2 ⊢ Ⅎyφ
2		nfa1 1788	. 2 ⊢ Ⅎx∀x(x = y → φ)
3		ax11v 2096	. . 3 ⊢ (x = y → (φ → ∀x(x = y → φ)))
4		sp 1747	. . . 4 ⊢ (∀x(x = y → φ) → (x = y → φ))
5	4	com12 27	. . 3 ⊢ (x = y → (∀x(x = y → φ) → φ))
6	3, 5	impbid 183	. 2 ⊢ (x = y → (φ ↔ ∀x(x = y → φ)))
7	1, 2, 6	cbvex 1985	1 ⊢ (∃xφ ↔ ∃y∀x(x = y → φ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  2exsb  2132