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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-hbext | Structured version Visualization version GIF version |
Description: Closed form of hbex 2324. (Contributed by BJ, 10-Oct-2019.) |
Ref | Expression |
---|---|
bj-hbext | ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa2 2174 | . . . 4 ⊢ Ⅎ𝑥∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) | |
2 | hbnt 2295 | . . . . . 6 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | |
3 | 2 | alimi 1819 | . . . . 5 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
4 | bj-hbalt 34600 | . . . . 5 ⊢ (∀𝑦(¬ 𝜑 → ∀𝑥 ¬ 𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)) |
6 | 1, 5 | alrimi 2211 | . . 3 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑)) |
7 | hbnt 2295 | . . 3 ⊢ (∀𝑥(∀𝑦 ¬ 𝜑 → ∀𝑥∀𝑦 ¬ 𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)) |
9 | df-ex 1788 | . . 3 ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑) | |
10 | 9 | bicomi 227 | . 2 ⊢ (¬ ∀𝑦 ¬ 𝜑 ↔ ∃𝑦𝜑) |
11 | 10 | albii 1827 | . 2 ⊢ (∀𝑥 ¬ ∀𝑦 ¬ 𝜑 ↔ ∀𝑥∃𝑦𝜑) |
12 | 8, 10, 11 | 3imtr3g 298 | 1 ⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1541 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-or 848 df-ex 1788 df-nf 1792 |
This theorem is referenced by: bj-nfext 34631 |
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