Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hadcoma | Structured version Visualization version GIF version |
Description: Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
hadcoma | ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xorcom 1459 | . . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑)) | |
2 | biid 250 | . . 3 ⊢ (𝜒 ↔ 𝜒) | |
3 | 1, 2 | xorbi12i 1469 | . 2 ⊢ (((𝜑 ⊻ 𝜓) ⊻ 𝜒) ↔ ((𝜓 ⊻ 𝜑) ⊻ 𝜒)) |
4 | df-had 1524 | . 2 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒)) | |
5 | df-had 1524 | . 2 ⊢ (hadd(𝜓, 𝜑, 𝜒) ↔ ((𝜓 ⊻ 𝜑) ⊻ 𝜒)) | |
6 | 3, 4, 5 | 3bitr4i 291 | 1 ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ⊻ wxo 1456 haddwhad 1523 |
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 196 df-xor 1457 df-had 1524 |
This theorem is referenced by: hadrot 1531 sadcom 15023 |
Copyright terms: Public domain | W3C validator |