Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 34831 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) Do not use 19.35 1880 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 34813 | . . . 4 ⊢ (∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓))) | |
2 | 1 | alimi 1814 | . . 3 ⊢ (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → ∀𝑥(∃𝑦𝜒 → (∀𝑦𝜑 → ∃𝑦𝜓))) |
3 | bj-alexim 34808 | . . 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 1836 | . . 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 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: bj-cbvexim 34827 |
Copyright terms: Public domain | W3C validator |