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Mirrors > Home > MPE Home > Th. List > xorbi12d | Structured version Visualization version GIF version |
Description: Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
xor12d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
xor12d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Ref | Expression |
---|---|
xorbi12d | ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xor12d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | xor12d.2 | . . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
3 | 1, 2 | bibi12d 345 | . . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜏))) |
4 | 3 | notbid 317 | . 2 ⊢ (𝜑 → (¬ (𝜓 ↔ 𝜃) ↔ ¬ (𝜒 ↔ 𝜏))) |
5 | df-xor 1506 | . 2 ⊢ ((𝜓 ⊻ 𝜃) ↔ ¬ (𝜓 ↔ 𝜃)) | |
6 | df-xor 1506 | . 2 ⊢ ((𝜒 ⊻ 𝜏) ↔ ¬ (𝜒 ↔ 𝜏)) | |
7 | 4, 5, 6 | 3bitr4g 313 | 1 ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ⊻ wxo 1505 |
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 206 df-xor 1506 |
This theorem is referenced by: hadbi123d 1599 cadbi123d 1615 |
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