Theorem sbtr 2421

Description: A partial converse to sbt 2419. 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	|- F/ y ph
sbtr.1	|- [ y / x ] ph

Assertion

Ref	Expression
sbtr	|- ph

Proof of Theorem sbtr

Step	Hyp	Ref	Expression
1		sbtr.nf	. . 3 |- F/ y ph
2	1	sbtrt 2420	. 2 |- ( A. y [ y / x ] ph -> ph )
3		sbtr.1	. 2 |- [ y / x ] ph
4	2, 3	mpg 1724	1 |- ph

Syntax hints:   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)