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Mirrors > Home > ILE Home > Th. List > xornbi | GIF version |
Description: A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1322. (Contributed by Jim Kingdon, 10-Mar-2018.) |
Ref | Expression |
---|---|
xornbi | ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xorbin 1315 | . 2 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓)) | |
2 | pm5.18im 1316 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓)) | |
3 | 2 | con2i 589 | . 2 ⊢ ((𝜑 ↔ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓)) |
4 | 1, 3 | syl 14 | 1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ⊻ wxo 1306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 df-xor 1307 |
This theorem is referenced by: (None) |
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