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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbv3tb | Structured version Visualization version GIF version |
Description: Closed form of cbv3 2406. (Contributed by BJ, 2-May-2019.) |
Ref | Expression |
---|---|
bj-cbv3tb | ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.9t 2194 | . . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓)) | |
2 | 1 | biimpd 230 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓)) |
3 | 2 | alimi 1803 | . 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦(∃𝑥𝜓 → 𝜓)) |
4 | nf5r 2183 | . . 3 ⊢ (Ⅎ𝑦𝜑 → (𝜑 → ∀𝑦𝜑)) | |
5 | 4 | alimi 1803 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ∀𝑥(𝜑 → ∀𝑦𝜑)) |
6 | bj-cbv3ta 34005 | . 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓))) | |
7 | 3, 5, 6 | syl2ani 606 | 1 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: (None) |
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