Metamath Proof Explorer


Theorem dfsb7

Description: An alternate definition of proper substitution df-sb . By introducing a dummy variable y in the definiens, we are able to eliminate any distinct variable restrictions among the variables t , x , and ph of the definiendum. No distinct variable conflicts arise because y effectively insulates t from x . To achieve this, we use a chain of two substitutions in the form of sb5 , first y for x then t for y . Compare Definition 2.1'' of Quine p. 17, which is obtained from this theorem by applying df-clab . Theorem sb7h provides a version where ph and y don't have to be distinct. (Contributed by NM, 28-Jan-2004) Revise df-sb . (Revised by BJ, 25-Dec-2020) (Proof shortened by Wolf Lammen, 3-Sep-2023)

Ref Expression
Assertion dfsb7 txφyy=txx=yφ

Proof

Step Hyp Ref Expression
1 sbalex yy=txx=yφyy=txx=yφ
2 sbalex xx=yφxx=yφ
3 2 anbi2i y=txx=yφy=txx=yφ
4 3 exbii yy=txx=yφyy=txx=yφ
5 df-sb txφyy=txx=yφ
6 1 4 5 3bitr4ri txφyy=txx=yφ