Theorem pm2.21fal 1363

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 610	1 ⊢ (𝜑 → ⊥)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in2 605

This theorem is referenced by:  genpdisj  7464  suplocexprlemdisj  7661  suplocexprlemub  7664  suplocsrlem  7749  recvguniqlem  10936  resqrexlemoverl  10963  leabs  11016  climge0  11266  isprm5lem  12073  dedekindeulemeu  13240  dedekindicclemeu  13249  pw1nct  13883