Metamath Proof Explorer


Theorem axac3

Description: This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 as an axiom. (Contributed by Mario Carneiro, 6-May-2015) (Revised by NM, 20-Dec-2016) (Proof modification is discouraged.)

Ref Expression
Assertion axac3 CHOICE

Proof

Step Hyp Ref Expression
1 ax-ac2 yzwvyxzywx¬y=wzw¬yxzxwzwyvzvyv=w
2 1 ax-gen xyzwvyxzywx¬y=wzw¬yxzxwzwyvzvyv=w
3 dfackm CHOICExyzwvyxzywx¬y=wzw¬yxzxwzwyvzvyv=w
4 2 3 mpbir CHOICE