Theorem notnotri 126

Description: Inference associated with notnotr 125.

Remark: the proof via notnotr 125 and ax-mp 5 also has three essential steps, but has a total number of steps equal to 8, instead of the present 7, because it has to construct the formula ph twice and the formula -. -. ph, whereas the present proof has to construct the formula ph twice and the formula -. ph, and therefore makes only one use of wn 3 instead of two. This can be checked by running the Metamath command "SHOW PROOF notnotri / NORMAL". (Contributed by NM, 27-Feb-2008.) (Proof shortened by Wolf Lammen, 15-Jul-2021.)

Hypothesis

Ref	Expression
notnotri.1	|- -. -. ph

Assertion

Ref	Expression
notnotri	|- ph

Proof of Theorem notnotri

Step	Hyp	Ref	Expression
1		notnotri.1	. . 3 |- -. -. ph
2	1	pm2.21i 116	. 2 |- ( -. ph -> ph )
3	2	pm2.18i 123	1 |- ph

Syntax hints:   -. wn 3

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  mt3  192  pm2.65ni  39210