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Mirrors > Home > MPE Home > Th. List > trut | Structured version Visualization version GIF version |
Description: A proposition is equivalent to it being implied by ⊤. Closed form of mptru 1535. Dual of dfnot 1547. It is to tbtru 1536 what a1bi 364 is to tbt 371. (Contributed by BJ, 26-Oct-2019.) |
Ref | Expression |
---|---|
trut | ⊢ (𝜑 ↔ (⊤ → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1532 | . 2 ⊢ ⊤ | |
2 | 1 | a1bi 364 | 1 ⊢ (𝜑 ↔ (⊤ → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ⊤wtru 1529 |
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 208 df-tru 1531 |
This theorem is referenced by: truimfal 1552 euae 2740 |
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