Theorem nic-isw2

Description: Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	nic-isw2.1	⊢ ( 𝜓 ⊼ ( 𝜃 ⊼ 𝜑 ) )
Assertion	nic-isw2	⊢ ( 𝜓 ⊼ ( 𝜑 ⊼ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		nic-isw2.1	⊢ ( 𝜓 ⊼ ( 𝜃 ⊼ 𝜑 ) )
2		nic-swap	⊢ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( ( 𝜃 ⊼ 𝜑 ) ⊼ ( 𝜃 ⊼ 𝜑 ) ) )
3	2	nic-imp	⊢ ( ( 𝜓 ⊼ ( 𝜃 ⊼ 𝜑 ) ) ⊼ ( ( ( 𝜑 ⊼ 𝜃 ) ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ 𝜓 ) ) )
4	1 3	nic-mp	⊢ ( ( 𝜑 ⊼ 𝜃 ) ⊼ 𝜓 )
5	4	nic-isw1	⊢ ( 𝜓 ⊼ ( 𝜑 ⊼ 𝜃 ) )