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Mirrors > Home > ILE Home > Th. List > df-cdeq | GIF version |
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.) |
Ref | Expression |
---|---|
df-cdeq | ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | 1, 2, 3 | wcdeq 2812 | . 2 wff CondEq(𝑥 = 𝑦 → 𝜑) |
5 | 2, 3 | weq 1435 | . . 3 wff 𝑥 = 𝑦 |
6 | 5, 1 | wi 4 | . 2 wff (𝑥 = 𝑦 → 𝜑) |
7 | 4, 6 | wb 103 | 1 wff (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff set class |
This definition is referenced by: cdeqi 2814 cdeqri 2815 bdcdeq 11168 |
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