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  7891  suplocexprlemdisj  8088  suplocexprlemub  8091  suplocsrlem  8176  recvguniqlem  11776  resqrexlemoverl  11803  leabs  11856  climge0  12110  isprm5lem  12939  dedekindeulemeu  15814  dedekindicclemeu  15823  usgr1vr  16655  pw1nct  17199