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Mirrors > Home > ILE Home > Th. List > truxorfal | GIF version |
Description: A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Ref | Expression |
---|---|
truxorfal | ⊢ ((⊤ ⊻ ⊥) ↔ ⊤) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xor 1386 | . 2 ⊢ ((⊤ ⊻ ⊥) ↔ ((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥))) | |
2 | truorfal 1416 | . . 3 ⊢ ((⊤ ∨ ⊥) ↔ ⊤) | |
3 | notfal 1424 | . . . 4 ⊢ (¬ ⊥ ↔ ⊤) | |
4 | truan 1380 | . . . 4 ⊢ ((⊤ ∧ ⊥) ↔ ⊥) | |
5 | 3, 4 | xchnxbir 682 | . . 3 ⊢ (¬ (⊤ ∧ ⊥) ↔ ⊤) |
6 | 2, 5 | anbi12i 460 | . 2 ⊢ (((⊤ ∨ ⊥) ∧ ¬ (⊤ ∧ ⊥)) ↔ (⊤ ∧ ⊤)) |
7 | anidm 396 | . 2 ⊢ ((⊤ ∧ ⊤) ↔ ⊤) | |
8 | 1, 6, 7 | 3bitri 206 | 1 ⊢ ((⊤ ⊻ ⊥) ↔ ⊤) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∨ wo 709 ⊤wtru 1364 ⊥wfal 1368 ⊻ wxo 1385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-fal 1369 df-xor 1386 |
This theorem is referenced by: (None) |
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