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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-3xorbi123i | Structured version Visualization version GIF version |
Description: Equivalence theorem for triple xor. Copy of hadbi123i 1598. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
wl-3xorbii.1 | ⊢ (𝜓 ↔ 𝜒) |
wl-3xorbii.2 | ⊢ (𝜃 ↔ 𝜏) |
wl-3xorbii.3 | ⊢ (𝜂 ↔ 𝜁) |
Ref | Expression |
---|---|
wl-3xorbi123i | ⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-3xorbii.1 | . . . 4 ⊢ (𝜓 ↔ 𝜒) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒)) |
3 | wl-3xorbii.2 | . . . 4 ⊢ (𝜃 ↔ 𝜏) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → (𝜃 ↔ 𝜏)) |
5 | wl-3xorbii.3 | . . . 4 ⊢ (𝜂 ↔ 𝜁) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → (𝜂 ↔ 𝜁)) |
7 | 2, 4, 6 | wl-3xorbi123d 35157 | . 2 ⊢ (⊤ → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁))) |
8 | 7 | mptru 1546 | 1 ⊢ (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ⊤wtru 1540 haddwhad 1595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-ifp 1060 df-xor 1504 df-tru 1542 df-had 1596 |
This theorem is referenced by: (None) |
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