Metamath Proof Explorer

Theorem sbtr

Description: A partial converse to sbt . If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by BJ, 15-Sep-2018) (New usage is discouraged.)

Ref Expression
Hypotheses y φ
sbtr.1 y x φ
Assertion sbtr φ


Step Hyp Ref Expression
1 y φ
2 sbtr.1 y x φ
3 1 sbtrt y y x φ φ
4 3 2 mpg φ