Theorem exsb 1925

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		ax-17 1459	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	sb8eh 1776	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3		sb6 1807	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	3	exbii 1536	. 2 ⊢ (∃𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
5	2, 4	bitri 182	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282  ∃wex 1421  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  2exsb  1926