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Mirrors > Home > ILE Home > Th. List > alrot4 | GIF version |
Description: Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
Ref | Expression |
---|---|
alrot4 | ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alrot3 1465 | . . 3 ⊢ (∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑦𝜑) | |
2 | 1 | albii 1450 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑥∀𝑧∀𝑤∀𝑦𝜑) |
3 | alcom 1458 | . 2 ⊢ (∀𝑥∀𝑧∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑥∀𝑤∀𝑦𝜑) | |
4 | alcom 1458 | . . 3 ⊢ (∀𝑥∀𝑤∀𝑦𝜑 ↔ ∀𝑤∀𝑥∀𝑦𝜑) | |
5 | 4 | albii 1450 | . 2 ⊢ (∀𝑧∀𝑥∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) |
6 | 2, 3, 5 | 3bitri 205 | 1 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-7 1428 ax-gen 1429 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: fun11 5238 |
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