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Mirrors > Home > MPE Home > Th. List > pm13.18OLD | Structured version Visualization version GIF version |
Description: Obsolete version of pm13.18 3097 as of 14-May-2023. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pm13.18OLD | ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2825 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | 1 | biimprd 250 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐵 = 𝐶 → 𝐴 = 𝐶)) |
3 | 2 | necon3d 3037 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 → 𝐵 ≠ 𝐶)) |
4 | 3 | imp 409 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ≠ wne 3016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-ne 3017 |
This theorem is referenced by: (None) |
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