Theorem imnani 412

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 411	. 2 ⊢ ((φ → ¬ ψ) ↔ ¬ (φ ∧ ψ))
3	1, 2	mpbir 200	1 ⊢ (φ → ¬ ψ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  mpto1  1533  eueq3  3012