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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 3101 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝜓) |
7 | 1, 6 | pm2.61dane 3101 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ≠ wne 3013 |
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 3014 |
This theorem is referenced by: pm2.61da3ne 3103 isabvd 19520 xrsxmet 23344 chordthmlem3 25339 mumul 25685 lgsdirnn0 25847 lgsdinn0 25848 lfl1dim 36137 lfl1dim2N 36138 pmodlem2 36863 cdlemg29 37721 cdlemg39 37732 cdlemg44b 37748 dia2dimlem9 38088 dihprrn 38442 dvh3dim 38462 lcfl9a 38521 lclkrlem2l 38534 lcfrlem42 38600 mapdh6kN 38762 hdmap1l6k 38836 |
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