Theorem mo3 2235

Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in φ in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)

Hypothesis

Ref	Expression
mo3.1	⊢ Ⅎyφ

Assertion

Ref	Expression
mo3	⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem mo3

Step	Hyp	Ref	Expression
1		mo3.1	. . 3 ⊢ Ⅎyφ
2	1	mo2 2233	. 2 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
3	1	mo 2226	. 2 ⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
4	2, 3	bitri 240	1 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648  ∃*wmo 2205

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  df-mo 2209

This theorem is referenced by:  mo4f  2236  mopick  2266  rmo3  3134