Metamath Proof Explorer


Theorem sb5f

Description: Equivalence for substitution when y is not free in ph . The implication "to the right" is sb1 and does not require the nonfreeness hypothesis. Theorem sb5 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, 5-Aug-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypothesis sb6f.1 y φ
Assertion sb5f y x φ x x = y φ

Proof

Step Hyp Ref Expression
1 sb6f.1 y φ
2 1 sb6f y x φ x x = y φ
3 1 equs45f x x = y φ x x = y φ
4 2 3 bitr4i y x φ x x = y φ