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Mirrors > Home > ILE Home > Th. List > necon2bbiddc | GIF version |
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.) |
Ref | Expression |
---|---|
necon2bbiddc.1 | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵))) |
Ref | Expression |
---|---|
necon2bbiddc | ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2bbiddc.1 | . . . 4 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵))) | |
2 | bicom 138 | . . . 4 ⊢ ((𝜓 ↔ 𝐴 ≠ 𝐵) ↔ (𝐴 ≠ 𝐵 ↔ 𝜓)) | |
3 | 1, 2 | syl6ib 159 | . . 3 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ 𝜓))) |
4 | 3 | necon1bbiddc 2308 | . 2 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 = 𝐵))) |
5 | bicom 138 | . 2 ⊢ ((¬ 𝜓 ↔ 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ↔ ¬ 𝜓)) | |
6 | 4, 5 | syl6ib 159 | 1 ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 DECID wdc 775 = wceq 1284 ≠ wne 2245 |
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-dc 776 df-ne 2246 |
This theorem is referenced by: (None) |
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