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	⊢ (𝜑 → 𝜓)
pm2.21fal.2	⊢ (𝜑 → ¬ 𝜓)

Assertion

Ref	Expression
pm2.21fal	⊢ (𝜑 → ⊥)

Proof of Theorem pm2.21fal

Step	Hyp	Ref	Expression
1		pm2.21fal.1	. 2 ⊢ (𝜑 → 𝜓)
2		pm2.21fal.2	. 2 ⊢ (𝜑 → ¬ 𝜓)
3	1, 2	pm2.21dd 629	1 ⊢ (𝜑 → ⊥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ⊥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  11762  resqrexlemoverl  11789  leabs  11842  climge0  12093  isprm5lem  12921  dedekindeulemeu  15725  dedekindicclemeu  15734  usgr1vr  16501  pw1nct  17045