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Mirrors > Home > MPE Home > Th. List > alrot4 | Structured version Visualization version GIF version |
Description: Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.) |
Ref | Expression |
---|---|
alrot4 | ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alrot3 2163 | . . 3 ⊢ (∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑦𝜑) | |
2 | 1 | albii 1827 | . 2 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑥∀𝑧∀𝑤∀𝑦𝜑) |
3 | alrot3 2163 | . 2 ⊢ (∀𝑥∀𝑧∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) | |
4 | 2, 3 | bitri 278 | 1 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-11 2160 |
This theorem depends on definitions: df-bi 210 |
This theorem is referenced by: 2mo 2649 fun11 6432 |
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