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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 2138. (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 1875 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | |
2 | aaan.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | 19.3 2196 | . . . 4 ⊢ (∀𝑦𝜑 ↔ 𝜑) |
4 | 3 | albii 1822 | . . 3 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥𝜑) |
5 | alcom 2157 | . . . 4 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦∀𝑥𝜓) | |
6 | aaan.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
7 | 6 | 19.3 2196 | . . . . 5 ⊢ (∀𝑥𝜓 ↔ 𝜓) |
8 | 7 | albii 1822 | . . . 4 ⊢ (∀𝑦∀𝑥𝜓 ↔ ∀𝑦𝜓) |
9 | 5, 8 | bitri 275 | . . 3 ⊢ (∀𝑥∀𝑦𝜓 ↔ ∀𝑦𝜓) |
10 | 4, 9 | anbi12i 628 | . 2 ⊢ ((∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
11 | 1, 10 | bitri 275 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∀wal 1540 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 |
This theorem is referenced by: aaanv 42742 pm11.71 42751 |
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