Metamath Proof Explorer

Theorem hbalgVD

Description: Virtual deduction proof of hbalg . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. hbalg is hbalgVD without virtual deductions and was automatically derived from hbalgVD . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

1:: |- (. A. y ( ph -> A. x ph ) ->. A. y ( ph -> A. x ph ) ).
2:1: |- (. A. y ( ph -> A. x ph ) ->. ( A. y ph -> A. y A. x ph ) ).
3:: |- ( A. y A. x ph -> A. x A. y ph )
4:2,3: |- (. A. y ( ph -> A. x ph ) ->. ( A. y ph -> A. x A. y ph ) ).
5:: |- ( A. y ( ph -> A. x ph ) -> A. y A. y ( ph -> A. x ph ) )
6:5,4: |- (. A. y ( ph -> A. x ph ) ->. A. y ( A. y ph -> A. x A. y ph ) ).
qed:6: |- ( A. y ( ph -> A. x ph ) -> A. y ( A. y ph -> A. x A. y ph ) )

Ref Expression
Assertion hbalgVD ( ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑦 ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )

Proof

Step Hyp Ref Expression
1 hba1 ⊢ ( ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑦 ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) )
2 idn1 ⊢ (    ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 )    ▶    ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 )    )
3 alim ⊢ ( ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) )
4 2 3 e1a ⊢ (    ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 )    ▶    ( ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )    )
5 ax-11 ⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )
6 imim1 ⊢ ( ( ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) → ( ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) ) )
7 4 5 6 e10 ⊢ (    ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 )    ▶    ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )    )
8 1 7 gen11nv ⊢ (    ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 )    ▶    ∀ 𝑦 ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )    )
9 8 in1 ⊢ ( ∀ 𝑦 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑦 ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 ) )