Theorem eu3 2230

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eu3.1	⊢ Ⅎyφ

Assertion

Ref	Expression
eu3	⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y)))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem eu3

Step	Hyp	Ref	Expression
1		eu3.1	. . 3 ⊢ Ⅎyφ
2	1	eu2 2229	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))
3	1	mo 2226	. . 3 ⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
4	3	anbi2i 675	. 2 ⊢ ((∃xφ ∧ ∃y∀x(φ → x = y)) ↔ (∃xφ ∧ ∀x∀y((φ ∧ [y / x]φ) → x = y)))
5	2, 4	bitr4i 243	1 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃y∀x(φ → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648  ∃!weu 2204

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  mo2  2233  eu5  2242  2eu4  2287  eqeu  3008  reu3  3027