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Mirrors > Home > MPE Home > Th. List > xorexmid | Structured version Visualization version GIF version |
Description: Exclusive-or variant of the law of the excluded middle (exmid 431). This statement is ancient, going back to at least Stoic logic. This statement does not necessarily hold in intuitionistic logic. (Contributed by David A. Wheeler, 23-Feb-2019.) |
Ref | Expression |
---|---|
xorexmid | ⊢ (𝜑 ⊻ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 375 | . 2 ⊢ ¬ (𝜑 ↔ ¬ 𝜑) | |
2 | df-xor 1465 | . 2 ⊢ ((𝜑 ⊻ ¬ 𝜑) ↔ ¬ (𝜑 ↔ ¬ 𝜑)) | |
3 | 1, 2 | mpbir 221 | 1 ⊢ (𝜑 ⊻ ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ⊻ wxo 1464 |
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 197 df-xor 1465 |
This theorem is referenced by: (None) |
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