Metamath Proof Explorer


Theorem sb6f

Description: Equivalence for substitution when y is not free in ph . The implication "to the left" is sb2 and does not require the nonfreeness hypothesis. Theorem sb6 replaces the nonfreeness hypothesis with a disjoint variable condition on x , y and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypothesis sb6f.1 yφ
Assertion sb6f yxφxx=yφ

Proof

Step Hyp Ref Expression
1 sb6f.1 yφ
2 1 nf5ri φyφ
3 2 sbimi yxφyxyφ
4 sb4a yxyφxx=yφ
5 3 4 syl yxφxx=yφ
6 sb2 xx=yφyxφ
7 5 6 impbii yxφxx=yφ