Theorem ifpdfan2

Description: Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020)

	Ref	Expression
Assertion	ifpdfan2	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝜑 → 𝜑 )
2	1	notnoti	⊢ ¬ ¬ ( 𝜑 → 𝜑 )
3	2	biorfri	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜑 → 𝜑 ) ) )
4		dfifp6	⊢ ( if- ( 𝜑 , 𝜓 , 𝜑 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∨ ¬ ( 𝜑 → 𝜑 ) ) )
5	3 4	bitr4i	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , 𝜑 ) )