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Mirrors > Home > ILE Home > Th. List > falxorfal | GIF version |
Description: A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Ref | Expression |
---|---|
falxorfal | ⊢ ((⊥ ⊻ ⊥) ↔ ⊥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xor 1376 | . 2 ⊢ ((⊥ ⊻ ⊥) ↔ ((⊥ ∨ ⊥) ∧ ¬ (⊥ ∧ ⊥))) | |
2 | oridm 757 | . . 3 ⊢ ((⊥ ∨ ⊥) ↔ ⊥) | |
3 | notfal 1414 | . . . 4 ⊢ (¬ ⊥ ↔ ⊤) | |
4 | anidm 396 | . . . 4 ⊢ ((⊥ ∧ ⊥) ↔ ⊥) | |
5 | 3, 4 | xchnxbir 681 | . . 3 ⊢ (¬ (⊥ ∧ ⊥) ↔ ⊤) |
6 | 2, 5 | anbi12i 460 | . 2 ⊢ (((⊥ ∨ ⊥) ∧ ¬ (⊥ ∧ ⊥)) ↔ (⊥ ∧ ⊤)) |
7 | falantru 1403 | . 2 ⊢ ((⊥ ∧ ⊤) ↔ ⊥) | |
8 | 1, 6, 7 | 3bitri 206 | 1 ⊢ ((⊥ ⊻ ⊥) ↔ ⊥) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∨ wo 708 ⊤wtru 1354 ⊥wfal 1358 ⊻ wxo 1375 |
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 614 ax-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-xor 1376 |
This theorem is referenced by: (None) |
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