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