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Mirrors > Home > ILE Home > Th. List > necon1bidc | GIF version |
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.) |
Ref | Expression |
---|---|
necon1bidc.1 | ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑)) |
Ref | Expression |
---|---|
necon1bidc | ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2328 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon1bidc.1 | . . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜑)) | |
3 | 1, 2 | syl5bir 152 | . 2 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵 → 𝜑)) |
4 | con1dc 842 | . 2 ⊢ (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵 → 𝜑) → (¬ 𝜑 → 𝐴 = 𝐵))) | |
5 | 3, 4 | mpd 13 | 1 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝜑 → 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 820 = wceq 1335 ≠ wne 2327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-ne 2328 |
This theorem is referenced by: (None) |
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