Theorem nic-bi2

Description: Inference to extract the other side of an implication from a 'biconditional' definition. (Contributed by Jeff Hoffman, 18-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	nic-bi2.1	⊢ ( ( 𝜑 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜓 ⊼ 𝜓 ) ) )
Assertion	nic-bi2	⊢ ( 𝜓 ⊼ ( 𝜑 ⊼ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nic-bi2.1	⊢ ( ( 𝜑 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜑 ) ⊼ ( 𝜓 ⊼ 𝜓 ) ) )
2	1	nic-isw2	⊢ ( ( 𝜑 ⊼ 𝜓 ) ⊼ ( ( 𝜓 ⊼ 𝜓 ) ⊼ ( 𝜑 ⊼ 𝜑 ) ) )
3		nic-id	⊢ ( 𝜓 ⊼ ( 𝜓 ⊼ 𝜓 ) )
4	2 3	nic-iimp1	⊢ ( 𝜓 ⊼ ( 𝜑 ⊼ 𝜓 ) )
5	4	nic-idel	⊢ ( 𝜓 ⊼ ( 𝜑 ⊼ 𝜑 ) )