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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-gl4lem | Structured version Visualization version GIF version |
Description: Lemma for bj-gl4 31753. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-gl4lem | ⊢ (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1786 | . . 3 ⊢ (∀𝑥(∀𝑥𝜑 ∧ 𝜑) ↔ (∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑)) | |
2 | simpr 476 | . . . . 5 ⊢ ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑)) |
4 | 3 | anc2ri 579 | . . 3 ⊢ (𝜑 → ((∀𝑥∀𝑥𝜑 ∧ ∀𝑥𝜑) → (∀𝑥𝜑 ∧ 𝜑))) |
5 | 1, 4 | syl5bi 231 | . 2 ⊢ (𝜑 → (∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑))) |
6 | 5 | alimi 1730 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑 ∧ 𝜑) → (∀𝑥𝜑 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀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 |
This theorem depends on definitions: df-bi 196 df-an 385 |
This theorem is referenced by: bj-gl4 31753 |
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