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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpdfan | Structured version Visualization version GIF version |
Description: Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.) |
Ref | Expression |
---|---|
ifpdfan | ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fal 1557 | . . . 4 ⊢ ¬ ⊥ | |
2 | 1 | intnan 490 | . . 3 ⊢ ¬ (¬ 𝜑 ∧ ⊥) |
3 | 2 | biorfi 939 | . 2 ⊢ ((𝜑 ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ⊥))) |
4 | df-ifp 1064 | . 2 ⊢ (if-(𝜑, 𝜓, ⊥) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ⊥))) | |
5 | 3, 4 | bitr4i 281 | 1 ⊢ ((𝜑 ∧ 𝜓) ↔ if-(𝜑, 𝜓, ⊥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 847 if-wif 1063 ⊥wfal 1555 |
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 210 df-an 400 df-or 848 df-ifp 1064 df-tru 1546 df-fal 1556 |
This theorem is referenced by: ifpdfnan 40778 ifpdfxor 40779 |
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