Theorem imnani 691

Description: Express implication in terms of 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 690	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	1, 2	mpbir 146	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  mptnan  1423  eueq3dc  2913  dtruex  4560  canth  5831  nntri2  6497  nndcel  6503