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Theorem 3impexpVD 39405
Description: Virtual deduction proof of 3impexp 1311. 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.
 1:: ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ) 2:: ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) 3:1,2,?: e10 39236 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)   ) 4:3,?: e1a 39169 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))   ) 5:4,?: e1a 39169 ⊢ (   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ▶   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ) 6:5: ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒 → 𝜃)))) 7:: ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ) 8:7,?: e1a 39169 ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))   ) 9:8,?: e1a 39169 ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)   ) 10:2,9,?: e01 39233 ⊢ (   (𝜑 → (𝜓 → (𝜒 → 𝜃)))   ▶   ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)   ) 11:10: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)) qed:6,11,?: e00 39312 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
3impexpVD (((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Proof of Theorem 3impexpVD
StepHypRef Expression
1 idn1 39107 . . . . . 6 (   ((𝜑𝜓𝜒) → 𝜃)   ▶   ((𝜑𝜓𝜒) → 𝜃)   )
2 df-3an 1056 . . . . . 6 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
3 imbi1 336 . . . . . . 7 (((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒)) → (((𝜑𝜓𝜒) → 𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) → 𝜃)))
43biimpcd 239 . . . . . 6 (((𝜑𝜓𝜒) → 𝜃) → (((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒)) → (((𝜑𝜓) ∧ 𝜒) → 𝜃)))
51, 2, 4e10 39236 . . . . 5 (   ((𝜑𝜓𝜒) → 𝜃)   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
6 pm3.3 459 . . . . 5 ((((𝜑𝜓) ∧ 𝜒) → 𝜃) → ((𝜑𝜓) → (𝜒𝜃)))
75, 6e1a 39169 . . . 4 (   ((𝜑𝜓𝜒) → 𝜃)   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
8 pm3.3 459 . . . 4 (((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜒𝜃))))
97, 8e1a 39169 . . 3 (   ((𝜑𝜓𝜒) → 𝜃)   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
109in1 39104 . 2 (((𝜑𝜓𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒𝜃))))
11 idn1 39107 . . . . . 6 (   (𝜑 → (𝜓 → (𝜒𝜃)))   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
12 pm3.31 460 . . . . . 6 ((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜑𝜓) → (𝜒𝜃)))
1311, 12e1a 39169 . . . . 5 (   (𝜑 → (𝜓 → (𝜒𝜃)))   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
14 pm3.31 460 . . . . 5 (((𝜑𝜓) → (𝜒𝜃)) → (((𝜑𝜓) ∧ 𝜒) → 𝜃))
1513, 14e1a 39169 . . . 4 (   (𝜑 → (𝜓 → (𝜒𝜃)))   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
163biimprd 238 . . . 4 (((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒)) → ((((𝜑𝜓) ∧ 𝜒) → 𝜃) → ((𝜑𝜓𝜒) → 𝜃)))
172, 15, 16e01 39233 . . 3 (   (𝜑 → (𝜓 → (𝜒𝜃)))   ▶   ((𝜑𝜓𝜒) → 𝜃)   )
1817in1 39104 . 2 ((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜑𝜓𝜒) → 𝜃))
19 impbi 198 . 2 ((((𝜑𝜓𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒𝜃)))) → (((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜑𝜓𝜒) → 𝜃)) → (((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))))
2010, 18, 19e00 39312 1 (((𝜑𝜓𝜒) → 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054 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 197  df-an 385  df-3an 1056  df-vd1 39103 This theorem is referenced by: (None)
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