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Mirrors > Home > MPE Home > Th. List > nic-bi2 | Structured version Visualization version GIF version |
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 |
---|---|
nic-bi2.1 | ⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) |
Ref | Expression |
---|---|
nic-bi2 | ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nic-bi2.1 | . . . 4 ⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) | |
2 | 1 | nic-isw2 1597 | . . 3 ⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜓 ⊼ 𝜓) ⊼ (𝜑 ⊼ 𝜑))) |
3 | nic-id 1594 | . . 3 ⊢ (𝜓 ⊼ (𝜓 ⊼ 𝜓)) | |
4 | 2, 3 | nic-iimp1 1598 | . 2 ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜓)) |
5 | 4 | nic-idel 1600 | 1 ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ⊼ wnan 1439 |
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 196 df-an 385 df-nan 1440 |
This theorem is referenced by: nic-stdmp 1606 nic-luk1 1607 nic-luk2 1608 nic-luk3 1609 |
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