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Theorem nic-bi2 1762
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 ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))
Assertion
Ref Expression
nic-bi2 (𝜓 ⊼ (𝜑𝜑))

Proof of Theorem nic-bi2
StepHypRef Expression
1 nic-bi2.1 . . . 4 ((𝜑𝜓) ⊼ ((𝜑𝜑) ⊼ (𝜓𝜓)))
21nic-isw2 1754 . . 3 ((𝜑𝜓) ⊼ ((𝜓𝜓) ⊼ (𝜑𝜑)))
3 nic-id 1751 . . 3 (𝜓 ⊼ (𝜓𝜓))
42, 3nic-iimp1 1755 . 2 (𝜓 ⊼ (𝜑𝜓))
54nic-idel 1757 1 (𝜓 ⊼ (𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wnan 1595
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 197  df-an 383  df-nan 1596
This theorem is referenced by:  nic-stdmp  1763  nic-luk1  1764  nic-luk2  1765  nic-luk3  1766
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