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Mirrors > Home > MPE Home > Th. List > aaan | Structured version Visualization version GIF version |
Description: Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2140. (Revised by Gino Giotto, 21-Nov-2024.) |
Ref | Expression |
---|---|
aaan.1 | ⊢ Ⅎ𝑦𝜑 |
aaan.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
aaan | ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26-2 1877 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | |
2 | aaan.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | 19.3 2198 | . . . 4 ⊢ (∀𝑦𝜑 ↔ 𝜑) |
4 | 3 | albii 1825 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥𝜑) |
5 | alcom 2159 | . . . 4 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦∀𝑥𝜓) | |
6 | aaan.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
7 | 6 | 19.3 2198 | . . . . 5 ⊢ (∀𝑥𝜓 ↔ 𝜓) |
8 | 7 | albii 1825 | . . . 4 ⊢ (∀𝑦∀𝑥𝜓 ↔ ∀𝑦𝜓) |
9 | 5, 8 | bitri 274 | . . 3 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦𝜓) |
10 | 4, 9 | anbi12i 626 | . 2 ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
11 | 1, 10 | bitri 274 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∀wal 1539 Ⅎwnf 1789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-11 2157 ax-12 2174 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1786 df-nf 1790 |
This theorem is referenced by: aaanv 41959 pm11.71 41968 |
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