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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbalgVD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of hbalg 37792.
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 37792
is hbalgVD 38163 without virtual deductions and was automatically derived
from hbalgVD 38163. (Contributed by Alan Sare, 8-Feb-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
Ref | Expression |
---|---|
hbalgVD | ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 2137 | . . 3 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑦(𝜑 → ∀𝑥𝜑)) | |
2 | idn1 37811 | . . . . 5 ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦(𝜑 → ∀𝑥𝜑) ) | |
3 | alim 1729 | . . . . 5 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)) | |
4 | 2, 3 | e1a 37873 | . . . 4 ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) ) |
5 | ax-11 2021 | . . . 4 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) | |
6 | imim1 81 | . . . 4 ⊢ ((∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) → ((∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))) | |
7 | 4, 5, 6 | e10 37940 | . . 3 ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) ) |
8 | 1, 7 | gen11nv 37863 | . 2 ⊢ ( ∀𝑦(𝜑 → ∀𝑥𝜑) ▶ ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) ) |
9 | 8 | in1 37808 | 1 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 df-vd1 37807 |
This theorem is referenced by: (None) |
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