Theorem mo3 1027

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.

Hypothesis

Ref	Expression
mo3.1	⊢ (φ → ∀yφ)

Assertion

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

Distinct variable group(s):   x,y

Proof of Theorem mo3

Step	Hyp	Ref	Expression
1		mo3.1	. . 3 ⊢ (φ → ∀yφ)
2	1	mo2 1026	. 2 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
3	1	mo 1020	. 2 ⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
4	2, 3	bitr 151	1 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852  ∃*wmo 1008

This theorem is referenced by:  mo4f 1028  mopick 1054

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

metamath.org