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| Mirrors > Home > MPE Home > Th. List > Mathboxes > exbiriVD | Structured version Visualization version GIF version | ||
| Description: Virtual deduction proof of exbiri 811.  The following user's proof is
       completed by invoking mmj2's unify command and using mmj2's StepSelector
       to pick all remaining steps of the Metamath proof. 
 | 
| Ref | Expression | 
|---|---|
| exbiriVD.1 | ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | 
| Ref | Expression | 
|---|---|
| exbiriVD | ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idn3 44635 | . . . . 5 ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜃 ) | |
| 2 | idn2 44633 | . . . . . 6 ⊢ ( 𝜑 , 𝜓 ▶ 𝜓 ) | |
| 3 | idn1 44594 | . . . . . . 7 ⊢ ( 𝜑 ▶ 𝜑 ) | |
| 4 | exbiriVD.1 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) | |
| 5 | pm3.3 448 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))) | |
| 6 | 5 | com12 32 | . . . . . . 7 ⊢ (𝜑 → (((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) → (𝜓 → (𝜒 ↔ 𝜃)))) | 
| 7 | 3, 4, 6 | e10 44714 | . . . . . 6 ⊢ ( 𝜑 ▶ (𝜓 → (𝜒 ↔ 𝜃)) ) | 
| 8 | pm2.27 42 | . . . . . 6 ⊢ (𝜓 → ((𝜓 → (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃))) | |
| 9 | 2, 7, 8 | e21 44750 | . . . . 5 ⊢ ( 𝜑 , 𝜓 ▶ (𝜒 ↔ 𝜃) ) | 
| 10 | biimpr 220 | . . . . . 6 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 → 𝜒)) | |
| 11 | 10 | com12 32 | . . . . 5 ⊢ (𝜃 → ((𝜒 ↔ 𝜃) → 𝜒)) | 
| 12 | 1, 9, 11 | e32 44778 | . . . 4 ⊢ ( 𝜑 , 𝜓 , 𝜃 ▶ 𝜒 ) | 
| 13 | 12 | in3 44629 | . . 3 ⊢ ( 𝜑 , 𝜓 ▶ (𝜃 → 𝜒) ) | 
| 14 | 13 | in2 44625 | . 2 ⊢ ( 𝜑 ▶ (𝜓 → (𝜃 → 𝜒)) ) | 
| 15 | 14 | in1 44591 | 1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 | 
| 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-3an 1089 df-vd1 44590 df-vd2 44598 df-vd3 44610 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |