Theorem al2imVD 42482

Description: Virtual deduction proof of al2im 1817. 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 42247	⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))    ▶   (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒))   )
3::	⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
4:2,3,?: e10 42314	⊢ (   ∀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 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Proof of Theorem al2imVD

Step	Hyp	Ref	Expression
1		idn1 42194	. . . 4 ⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))   ▶   ∀𝑥(𝜑 → (𝜓 → 𝜒))   )
2		alim 1813	. . . 4 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)))
3	1, 2	e1a 42247	. . 3 ⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))   ▶   (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒))   )
4		alim 1813	. . 3 ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
5		imim1 83	. . 3 ⊢ ((∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) → ((∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))))
6	3, 4, 5	e10 42314	. 2 ⊢ (   ∀𝑥(𝜑 → (𝜓 → 𝜒))   ▶   (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))   )
7	6	in1 42191	1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-vd1 42190

This theorem is referenced by: (None)