MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-tru Structured version   Visualization version   GIF version

Definition df-tru 1542
Description: Definition of the truth value "true", or "verum", denoted by . In this definition, an instance of id 22 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular instance of id 22 was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1543, and other proofs should use tru 1543 instead of this definition, since there are many alternate ways to define . (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) Use tru 1543 instead. (New usage is discouraged.)
Assertion
Ref Expression
df-tru (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Detailed syntax breakdown of Definition df-tru
StepHypRef Expression
1 wtru 1540 . 2 wff
2 vx.tru . . . . . 6 setvar 𝑥
32cv 1538 . . . . 5 class 𝑥
43, 3wceq 1539 . . . 4 wff 𝑥 = 𝑥
54, 2wal 1537 . . 3 wff 𝑥 𝑥 = 𝑥
65, 5wi 4 . 2 wff (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)
71, 6wb 205 1 wff (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  tru  1543
  Copyright terms: Public domain W3C validator