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| Mirrors > Home > MPE Home > Th. List > pm2.61ii | Structured version Visualization version GIF version | ||
| Description: Inference eliminating two antecedents. (Contributed by NM, 4-Jan-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) | 
| Ref | Expression | 
|---|---|
| pm2.61ii.1 | ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒)) | 
| pm2.61ii.2 | ⊢ (𝜑 → 𝜒) | 
| pm2.61ii.3 | ⊢ (𝜓 → 𝜒) | 
| Ref | Expression | 
|---|---|
| pm2.61ii | ⊢ 𝜒 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pm2.61ii.2 | . 2 ⊢ (𝜑 → 𝜒) | |
| 2 | pm2.61ii.1 | . . 3 ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜒)) | |
| 3 | pm2.61ii.3 | . . 3 ⊢ (𝜓 → 𝜒) | |
| 4 | 2, 3 | pm2.61d2 181 | . 2 ⊢ (¬ 𝜑 → 𝜒) | 
| 5 | 1, 4 | pm2.61i 182 | 1 ⊢ 𝜒 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 | 
| This theorem is referenced by: pm2.61iii 185 hbae 2435 pssnn 9209 alephadd 10618 axextnd 10632 axunnd 10637 axpownd 10642 axregndlem2 10644 axregnd 10645 axinfndlem1 10646 axinfnd 10647 2cshwcshw 14865 ressress 17294 frgrreg 30414 bj-hbaeb2 36820 hbae-o 38905 hbequid 38911 ax5eq 38934 ax5el 38939 odd2prm2 47710 | 
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