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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 711 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 = 𝐷) → 𝜓) |
4 | pm2.61da2ne.3 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷)) → 𝜓) | |
5 | 4 | anassrs 468 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ 𝐵) ∧ 𝐶 ≠ 𝐷) → 𝜓) |
6 | 3, 5 | pm2.61dane 3104 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓) |
7 | 1, 6 | pm2.61dane 3104 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ≠ wne 3016 |
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 208 df-an 397 df-ne 3017 |
This theorem is referenced by: pm2.61da3ne 3106 isabvd 19522 xrsxmet 23346 chordthmlem3 25339 mumul 25686 lgsdirnn0 25848 lgsdinn0 25849 lfl1dim 36139 lfl1dim2N 36140 pmodlem2 36865 cdlemg29 37723 cdlemg39 37734 cdlemg44b 37750 dia2dimlem9 38090 dihprrn 38444 dvh3dim 38464 lcfl9a 38523 lclkrlem2l 38536 lcfrlem42 38602 mapdh6kN 38764 hdmap1l6k 38838 |
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