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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbveximt | Structured version Visualization version GIF version |
Description: A lemma in closed form used to prove bj-cbvex 34810 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1883 since only the direction of the biconditional used here holds in intuitionistic logic. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbveximt | ⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-exalim 34792 | . . . 4 ⊢ (∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓))) | |
2 | 1 | alimi 1817 | . . 3 ⊢ (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → ∀𝑥(∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓))) |
3 | bj-alexim 34787 | . . 3 ⊢ (∀𝑥(∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓)) → (∀𝑥∃𝑦𝜒 → (∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥∃𝑦𝜒 → (∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓))) |
5 | exim 1839 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∃𝑥∀𝑦𝜑)) | |
6 | imim2 58 | . . 3 ⊢ ((∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓) → ((∃𝑥𝜑 → ∃𝑥∀𝑦𝜑) → (∃𝑥𝜑 → ∃𝑥∃𝑦𝜓))) | |
7 | imim1 83 | . . 3 ⊢ ((∃𝑥𝜑 → ∃𝑥∃𝑦𝜓) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓))) | |
8 | 5, 6, 7 | syl56 36 | . 2 ⊢ ((∃𝑥∀𝑦𝜑 → ∃𝑥∃𝑦𝜓) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓)))) |
9 | 4, 8 | syl6com 37 | 1 ⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∀𝑥(𝜑 → ∀𝑦𝜑) → ((∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) → (∃𝑥𝜑 → ∃𝑦𝜓))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∃wex 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 |
This theorem depends on definitions: df-bi 206 df-ex 1786 |
This theorem is referenced by: bj-cbvexim 34806 |
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