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 y φ
Assertion sb6f y x φ x x = y φ

Proof

Step Hyp Ref Expression
1 sb6f.1 y φ
2 1 nf5ri φ y φ
3 2 sbimi y x φ y x y φ
4 sb4a y x y φ x x = y φ
5 3 4 syl y x φ x x = y φ
6 sb2 x x = y φ y x φ
7 5 6 impbii y x φ x x = y φ