Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > had1 | Structured version Visualization version GIF version |
Description: If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) |
Ref | Expression |
---|---|
had1 | ⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hadrot 1540 | . . . 4 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑)) | |
2 | hadbi 1537 | . . . 4 ⊢ (hadd(𝜓, 𝜒, 𝜑) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑)) | |
3 | 1, 2 | bitri 264 | . . 3 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑)) |
4 | biass 374 | . . 3 ⊢ (((hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)) ↔ 𝜑) ↔ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜓 ↔ 𝜒) ↔ 𝜑))) | |
5 | 3, 4 | mpbir 221 | . 2 ⊢ ((hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)) ↔ 𝜑) |
6 | 5 | biimpri 218 | 1 ⊢ (𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 haddwhad 1532 |
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 197 df-xor 1465 df-had 1533 |
This theorem is referenced by: had0 1543 hadifp 1544 sadadd2lem2 15172 |
Copyright terms: Public domain | W3C validator |