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| Mirrors > Home > MPE Home > Th. List > Mathboxes > e21 | Structured version Visualization version GIF version | ||
| Description: A virtual deduction elimination rule (see syl6ci 71). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| e21.1 | ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) | 
| e21.2 | ⊢ ( 𝜑 ▶ 𝜃 ) | 
| e21.3 | ⊢ (𝜒 → (𝜃 → 𝜏)) | 
| Ref | Expression | 
|---|---|
| e21 | ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | e21.1 | . 2 ⊢ ( 𝜑 , 𝜓 ▶ 𝜒 ) | |
| 2 | e21.2 | . . 3 ⊢ ( 𝜑 ▶ 𝜃 ) | |
| 3 | 2 | vd12 44625 | . 2 ⊢ ( 𝜑 , 𝜓 ▶ 𝜃 ) | 
| 4 | e21.3 | . 2 ⊢ (𝜒 → (𝜃 → 𝜏)) | |
| 5 | 1, 3, 4 | e22 44696 | 1 ⊢ ( 𝜑 , 𝜓 ▶ 𝜏 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ( wvd1 44594 ( wvd2 44602 | 
| 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 207 df-an 396 df-vd1 44595 df-vd2 44603 | 
| This theorem is referenced by: e21an 44756 en3lplem1VD 44868 exbiriVD 44879 syl5impVD 44888 sbcim2gVD 44900 onfrALTlem3VD 44912 onfrALTlem2VD 44914 hbimpgVD 44929 ax6e2eqVD 44932 vk15.4jVD 44939 | 
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