Theorem nanan 1523

Description: Conjunction in terms of alternative denial. (Contributed by Mario Carneiro, 9-May-2015.)

Assertion

Ref	Expression
nanan	⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ⊼ 𝜓))

Proof of Theorem nanan

Step	Hyp	Ref	Expression
1		df-nan 1522	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2	1	con2bii 360	1 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ⊼ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ⊼ wnan 1521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-nan 1522

This theorem is used by:  nannan  1527  nanass  1540