Theorem ifpdfan

Description: Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020)

	Ref	Expression
Assertion	ifpdfan	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ⊥ ) )

Proof

Step	Hyp	Ref	Expression
1		fal	⊢ ¬ ⊥
2	1	intnan	⊢ ¬ ( ¬ 𝜑 ∧ ⊥ )
3	2	biorfi	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ⊥ ) ) )
4		df-ifp	⊢ ( if- ( 𝜑 , 𝜓 , ⊥ ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ( ¬ 𝜑 ∧ ⊥ ) ) )
5	3 4	bitr4i	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ⊥ ) )