Metamath Proof Explorer

Theorem al2imVD

Description: Virtual deduction proof of al2im . 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:: |- (. A. x ( ph -> ( ps -> ch ) ) ->. A. x ( ph -> ( ps -> ch ) ) ).
2:1,?: e1a |- (. A. x ( ph -> ( ps -> ch ) ) ->. ( A. x ph -> A. x ( ps -> ch ) ) ).
3:: |- ( A. x ( ps -> ch ) -> ( A. x ps -> A. x ch ) )
4:2,3,?: e10 |- (. A. x ( ph -> ( ps -> ch ) ) ->. ( A. x ph -> ( A. x ps -> A. x ch ) ) ).
qed:4: |- ( A. x ( ph -> ( ps -> ch ) ) -> ( A. x ph -> ( A. x ps -> A. x ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion al2imVD ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) )    ▶    ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) )    )
2 alim ⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) )
3 1 2 e1a ⊢ (    ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) )    ▶    ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) )    )
4 alim ⊢ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )
5 imim1 ⊢ ( ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝜓 → 𝜒 ) ) → ( ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) ) ) )
6 3 4 5 e10 ⊢ (    ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) )    ▶    ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )    )
7 6 in1 ⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) ) )