Theorem nf5di 2119

Description: Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either F/ x ph (inference form) or ( ph -> F/ x ph ) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)

Hypothesis

Ref	Expression
nf5di.1	|- ( ph -> F/ x ph )

Assertion

Ref	Expression
nf5di	|- F/ x ph

Proof of Theorem nf5di

Step	Hyp	Ref	Expression
1		nf5di.1	. . . 4 |- ( ph -> F/ x ph )
2	1	nf5rd 2066	. . 3 |- ( ph -> ( ph -> A. x ph ) )
3	2	pm2.43i 52	. 2 |- ( ph -> A. x ph )
4	3	nf5i 2024	1 |- F/ x ph

Syntax hints:   -> wi 4   A.wal 1481   F/wnf 1708

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

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)