Theorem pm2.21fal 1422

Description: If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses

Ref	Expression
pm2.21fal.1	|- ( ph -> ps )
pm2.21fal.2	|- ( ph -> -. ps )

Assertion

Ref	Expression
pm2.21fal	|- ( ph -> F. )

Proof of Theorem pm2.21fal

Step	Hyp	Ref	Expression
1		pm2.21fal.1	. 2 |- ( ph -> ps )
2		pm2.21fal.2	. 2 |- ( ph -> -. ps )
3	1, 2	pm2.21dd 629	1 |- ( ph -> F. )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   F. wfal 1407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in2 624

This theorem is used by:  genpdisj  7890  suplocexprlemdisj  8087  suplocexprlemub  8090  suplocsrlem  8175  recvguniqlem  11760  resqrexlemoverl  11787  leabs  11840  climge0  12091  isprm5lem  12919  dedekindeulemeu  15723  dedekindicclemeu  15732  usgr1vr  16489  pw1nct  17033