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  3671  onuninsuci  7773  infn0  9191  sucprcreg  9496  elnotel  9506  alephsucdom  9973  pwfseq  10558  eirr  16114  mreexmrid  17549  dvferm1  25887  dvferm2  25889  dchrisumn0  27430  rpvmasum  27435  cvnsym  32234  ballotlem2  34457  bnj1224  34768  bnj1541  34823  bnj1311  34991  bj-imn3ani  36565