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Mirrors > Home > MPE Home > Th. List > pm2.61da2ne | Structured version Visualization version GIF version |
Description: Deduction eliminating two inequalities in an antecedent. (Contributed by NM, 29-May-2013.) |
Ref | Expression |
---|---|
pm2.61da2ne.1 | ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓) |
pm2.61da2ne.2 | ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓) |
pm2.61da2ne.3 | ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓) |
Ref | Expression |
---|---|
pm2.61da2ne | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.61da2ne.1 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜓) | |
2 | pm2.61da2ne.2 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝜓) | |
3 | 2 | adantlr 714 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 = 𝐷) → 𝜓) |
4 | pm2.61da2ne.3 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓) | |
5 | 4 | anassrs 469 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ≠ 𝐷) → 𝜓) |
6 | 3, 5 | pm2.61dane 3030 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓) |
7 | 1, 6 | pm2.61dane 3030 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ≠ wne 2941 |
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 2942 |
This theorem is referenced by: pm2.61da3ne 3032 isabvd 20428 xrsxmet 24325 chordthmlem3 26339 mumul 26685 lgsdirnn0 26847 lgsdinn0 26848 lfl1dim 37991 lfl1dim2N 37992 pmodlem2 38718 cdlemg29 39576 cdlemg39 39587 cdlemg44b 39603 dia2dimlem9 39943 dihprrn 40297 dvh3dim 40317 lcfl9a 40376 lclkrlem2l 40389 lcfrlem42 40455 mapdh6kN 40617 hdmap1l6k 40691 |
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