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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 2228 | . . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓)) |
3 | 2 | albii 1821 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓)) |
4 | aaan.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfal 2331 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
6 | 5 | 19.27 2227 | . 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
7 | 3, 6 | bitri 278 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∀wal 1536 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 |
This theorem is referenced by: aaanv 41092 pm11.71 41101 |
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