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Mirrors > Home > MPE Home > Th. List > mteqand | Structured version Visualization version GIF version |
Description: A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023.) |
Ref | Expression |
---|---|
mteqand.1 | ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
mteqand.2 | ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
mteqand | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mteqand.1 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷) | |
2 | 1 | neneqd 2945 | . . 3 ⊢ (𝜑 → ¬ 𝐶 = 𝐷) |
3 | mteqand.2 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷) | |
4 | 2, 3 | mtand 815 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
5 | 4 | neqned 2947 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ≠ wne 2940 |
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 206 df-an 398 df-ne 2941 |
This theorem is referenced by: isdrngd 20226 qsidomlem2 32274 zarcmplem 32519 imadrhmcl 40761 ricdrng1 40762 remul01 40919 remulinvcom 40944 prjspersym 40988 prjspreln0 40990 prjspner1 41007 flt0 41018 fltne 41025 |
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