Theorem nic-bi2 1454

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.)

Hypothesis

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

Assertion

Ref	Expression
nic-bi2	⊢ (ψ ⊼ (φ ⊼ φ))

Proof of Theorem nic-bi2

Step	Hyp	Ref	Expression
1		nic-bi2.1	. . . 4 ⊢ ((φ ⊼ ψ) ⊼ ((φ ⊼ φ) ⊼ (ψ ⊼ ψ)))
2	1	nic-isw2 1446	. . 3 ⊢ ((φ ⊼ ψ) ⊼ ((ψ ⊼ ψ) ⊼ (φ ⊼ φ)))
3		nic-id 1443	. . 3 ⊢ (ψ ⊼ (ψ ⊼ ψ))
4	2, 3	nic-iimp1 1447	. 2 ⊢ (ψ ⊼ (φ ⊼ ψ))
5	4	nic-idel 1449	1 ⊢ (ψ ⊼ (φ ⊼ φ))

Syntax hints:   ⊼ wnan 1287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-nan 1288

This theorem is referenced by:  nic-stdmp  1455  nic-luk1  1456  nic-luk2  1457  nic-luk3  1458