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Mirrors > Home > MPE Home > Th. List > aaan | Structured version Visualization version GIF version |
Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
Ref | Expression |
---|---|
aaan.1 | ⊢ Ⅎ𝑦𝜑 |
aaan.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
aaan | ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aaan.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | 19.28 2220 | . . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓)) |
3 | 2 | albii 1811 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓)) |
4 | aaan.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfal 2333 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
6 | 5 | 19.27 2219 | . 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
7 | 3, 6 | bitri 276 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∀wal 1526 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-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: aaanv 40597 pm11.71 40606 |
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