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Theorem al2imVD 39514
Description: Virtual deduction proof of al2im 1855. 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:: ⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))    ▶   ∀𝑥(𝜑 → (𝜓 → 𝜒))   ) 2:1,?: e1a 39271 ⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))    ▶   (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒))   ) 3:: ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒)) 4:2,3,?: e10 39338 ⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))    ▶   (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))   ) qed:4: ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
al2imVD (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Proof of Theorem al2imVD
StepHypRef Expression
1 idn1 39209 . . . 4 (   𝑥(𝜑 → (𝜓𝜒))   ▶   𝑥(𝜑 → (𝜓𝜒))   )
2 alim 1851 . . . 4 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → ∀𝑥(𝜓𝜒)))
31, 2e1a 39271 . . 3 (   𝑥(𝜑 → (𝜓𝜒))   ▶   (∀𝑥𝜑 → ∀𝑥(𝜓𝜒))   )
4 alim 1851 . . 3 (∀𝑥(𝜓𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
5 imim1 83 . . 3 ((∀𝑥𝜑 → ∀𝑥(𝜓𝜒)) → ((∀𝑥(𝜓𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))))
63, 4, 5e10 39338 . 2 (   𝑥(𝜑 → (𝜓𝜒))   ▶   (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))   )
76in1 39206 1 (∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1594 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1850 This theorem depends on definitions:  df-bi 197  df-vd1 39205 This theorem is referenced by: (None)
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