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

Definition df-cdeq 3385
Description: Define conditional equality. All the notation to the left of the is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
Assertion
Ref Expression
df-cdeq (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))

Detailed syntax breakdown of Definition df-cdeq
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
41, 2, 3wcdeq 3384 . 2 wff CondEq(𝑥 = 𝑦𝜑)
52, 3weq 1860 . . 3 wff 𝑥 = 𝑦
65, 1wi 4 . 2 wff (𝑥 = 𝑦𝜑)
74, 6wb 194 1 wff (CondEq(𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
This definition is referenced by:  cdeqi  3386  cdeqri  3387
  Copyright terms: Public domain W3C validator