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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:: (   ((𝜑𝜓𝜒) 𝜃)   ▶   ((𝜑𝜓𝜒) → 𝜃)   )
2:: ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
3:1,2,?: e10 39236 (   ((𝜑𝜓𝜒) 𝜃)   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
4:3,?: e1a 39169 (   ((𝜑𝜓𝜒) 𝜃)   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
5:4,?: e1a 39169 (   ((𝜑𝜓𝜒) 𝜃)   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
6:5: (((𝜑𝜓𝜒) → 𝜃) → (𝜑 → (𝜓 → (𝜒𝜃))))
7:: (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   (𝜑 → (𝜓 → (𝜒𝜃)))   )
8:7,?: e1a 39169 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   ((𝜑𝜓) → (𝜒𝜃))   )
9:8,?: e1a 39169 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   (((𝜑𝜓) ∧ 𝜒) → 𝜃)   )
10:2,9,?: e01 39233 (   (𝜑 → (𝜓 → (𝜒 𝜃)))   ▶   ((𝜑𝜓𝜒) → 𝜃)   )
11:10: ((𝜑 → (𝜓 → (𝜒 𝜃))) → ((𝜑𝜓𝜒) → 𝜃))
qed:6,11,?: e00 39312 (((𝜑𝜓𝜒) 𝜃) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))
(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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