Theorem ifpdfnan

Description: Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020)

	Ref	Expression
Assertion	ifpdfnan	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ if- ( 𝜑 , ¬ 𝜓 , ⊤ ) )

Proof

Step	Hyp	Ref	Expression
1		df-nan	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ ¬ ( 𝜑 ∧ 𝜓 ) )
2		ifpdfan	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ if- ( 𝜑 , 𝜓 , ⊥ ) )
3	2	notbii	⊢ ( ¬ ( 𝜑 ∧ 𝜓 ) ↔ ¬ if- ( 𝜑 , 𝜓 , ⊥ ) )
4		ifpnot23	⊢ ( ¬ if- ( 𝜑 , 𝜓 , ⊥ ) ↔ if- ( 𝜑 , ¬ 𝜓 , ¬ ⊥ ) )
5		notfal	⊢ ( ¬ ⊥ ↔ ⊤ )
6		ifpbi3	⊢ ( ( ¬ ⊥ ↔ ⊤ ) → ( if- ( 𝜑 , ¬ 𝜓 , ¬ ⊥ ) ↔ if- ( 𝜑 , ¬ 𝜓 , ⊤ ) ) )
7	5 6	ax-mp	⊢ ( if- ( 𝜑 , ¬ 𝜓 , ¬ ⊥ ) ↔ if- ( 𝜑 , ¬ 𝜓 , ⊤ ) )
8	4 7	bitri	⊢ ( ¬ if- ( 𝜑 , 𝜓 , ⊥ ) ↔ if- ( 𝜑 , ¬ 𝜓 , ⊤ ) )
9	1 3 8	3bitri	⊢ ( ( 𝜑 ⊼ 𝜓 ) ↔ if- ( 𝜑 , ¬ 𝜓 , ⊤ ) )