Metamath Proof Explorer


Theorem axextmo

Description: There exists at most one set with prescribed elements. Theorem 1.1 of BellMachover p. 462. (Contributed by NM, 30-Jun-1994) (Proof shortened by Wolf Lammen, 13-Nov-2019) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022)

Ref Expression
Hypothesis axextmo.1 xφ
Assertion axextmo *xyyxφ

Proof

Step Hyp Ref Expression
1 axextmo.1 xφ
2 biantr yxφyzφyxyz
3 2 alanimi yyxφyyzφyyxyz
4 ax-ext yyxyzx=z
5 3 4 syl yyxφyyzφx=z
6 5 gen2 xzyyxφyyzφx=z
7 nfv xyz
8 7 1 nfbi xyzφ
9 8 nfal xyyzφ
10 elequ2 x=zyxyz
11 10 bibi1d x=zyxφyzφ
12 11 albidv x=zyyxφyyzφ
13 9 12 mo4f *xyyxφxzyyxφyyzφx=z
14 6 13 mpbir *xyyxφ