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Mirrors > Home > MPE Home > Th. List > anxordi | Structured version Visualization version GIF version |
Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1016 does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.) |
Ref | Expression |
---|---|
anxordi | ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xordi 1016 | . 2 ⊢ ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) | |
2 | df-xor 1507 | . . 3 ⊢ ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒)) | |
3 | 2 | anbi2i 626 | . 2 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ∧ ¬ (𝜓 ↔ 𝜒))) |
4 | df-xor 1507 | . 2 ⊢ (((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) | |
5 | 1, 3, 4 | 3bitr4i 306 | 1 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ⊻ wxo 1506 |
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 210 df-an 400 df-xor 1507 |
This theorem is referenced by: (None) |
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