Theorem imnani 400

Description: Infer an implication from a negated conjunction. (Contributed by Mario Carneiro, 28-Sep-2015.)

Hypothesis

Ref	Expression
imnani.1	⊢ ¬ (𝜑 ∧ 𝜓)

Assertion

Ref	Expression
imnani	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem imnani

Step	Hyp	Ref	Expression
1		imnani.1	. 2 ⊢ ¬ (𝜑 ∧ 𝜓)
2		imnan 399	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	1, 2	mpbir 231	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395

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

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  mptnan  1768  eueq3  3694  onuninsuci  7835  infn0  9312  sucprcreg  9615  elnotel  9624  alephsucdom  10093  pwfseq  10678  eirr  16223  mreexmrid  17655  dvferm1  25941  dvferm2  25943  dchrisumn0  27484  rpvmasum  27489  cvnsym  32271  ballotlem2  34521  bnj1224  34832  bnj1541  34887  bnj1311  35055  bj-imn3ani  36605