Definition df-tru 1540

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 1541, and other proofs should use tru 1541 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 1541 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 1538	. 2 wff ⊤
2		vx.tru	. . . . . 6 setvar 𝑥
3	2	cv 1536	. . . . 5 class 𝑥
4	3, 3	wceq 1537	. . . 4 wff 𝑥 = 𝑥
5	4, 2	wal 1535	. . 3 wff ∀𝑥 𝑥 = 𝑥
6	5, 5	wi 4	. 2 wff (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)
7	1, 6	wb 208	1 wff (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))

This definition is referenced by:  tru  1541