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Mirrors > Home > ILE Home > Th. List > aaan | 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 1563 | . . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓)) |
3 | 2 | albii 1470 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓)) |
4 | aaan.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfal 1576 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
6 | 5 | 19.27 1561 | . 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
7 | 3, 6 | bitri 184 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∀wal 1351 Ⅎwnf 1460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-4 1510 |
This theorem depends on definitions: df-bi 117 df-nf 1461 |
This theorem is referenced by: (None) |
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