Theorem sbtr 2164

Description: A partial converse to sbt 2162. If the substitution of a variable for a non-free one in a wff gives a theorem, then the original wff is a theorem. (Contributed by BJ, 15-Sep-2018.)

Hypotheses

Ref	Expression
sbtr.nf
sbtr.1

Assertion

Ref	Expression
sbtr

Proof of Theorem sbtr

Step	Hyp	Ref	Expression
1		sbtr.nf	. . 3
2	1	sbtrt 2163	. 2
3		sbtr.1	. 2
4	2, 3	mpg 1620	1

Syntax hints:  F/wnf 1616  [wsb 1739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740