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Mirrors > Home > MPE Home > Th. List > Mathboxes > un0.1 | Structured version Visualization version GIF version |
Description: ⊤ is the constant true, a tautology (see df-tru 1486). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
un0.1.1 | ⊢ ( ⊤ ▶ 𝜑 ) |
un0.1.2 | ⊢ ( 𝜓 ▶ 𝜒 ) |
un0.1.3 | ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) |
Ref | Expression |
---|---|
un0.1 | ⊢ ( 𝜓 ▶ 𝜃 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un0.1.1 | . . . 4 ⊢ ( ⊤ ▶ 𝜑 ) | |
2 | 1 | in1 38787 | . . 3 ⊢ (⊤ → 𝜑) |
3 | un0.1.2 | . . . 4 ⊢ ( 𝜓 ▶ 𝜒 ) | |
4 | 3 | in1 38787 | . . 3 ⊢ (𝜓 → 𝜒) |
5 | un0.1.3 | . . . 4 ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) | |
6 | 5 | dfvd2ani 38799 | . . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃) |
7 | 2, 4, 6 | uun0.1 39005 | . 2 ⊢ (𝜓 → 𝜃) |
8 | 7 | dfvd1ir 38789 | 1 ⊢ ( 𝜓 ▶ 𝜃 ) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1484 ( wvd1 38785 ( wvhc2 38796 |
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 386 df-tru 1486 df-vd1 38786 df-vhc2 38797 |
This theorem is referenced by: sspwimpVD 39155 |
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