Definition df-tru 1486

Description: Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1487. 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 id 22 instance 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 1487, and other proofs should depend on tru 1487 (directly or indirectly) 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 1487 instead. (New usage is discouraged.)

Assertion

Ref	Expression
df-tru	⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

Detailed syntax breakdown of Definition df-tru

Step	Hyp	Ref	Expression
1		wtru 1484	. 2 wff ⊤
2		vx.tru	. . . . . 6 setvar 𝑥
3	2	cv 1482	. . . . 5 class 𝑥
4	3, 3	wceq 1483	. . . 4 wff 𝑥 = 𝑥
5	4, 2	wal 1481	. . 3 wff ∀𝑥 𝑥 = 𝑥
6	5, 5	wi 4	. 2 wff (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)
7	1, 6	wb 196	1 wff (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

This definition is referenced by:  tru  1487