Theorem alrot4 1486

Description: Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.)

Assertion

Ref	Expression
alrot4	⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)

Proof of Theorem alrot4

Step	Hyp	Ref	Expression
1		alrot3 1485	. . 3 ⊢ (∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑦𝜑)
2	1	albii 1470	. 2 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑥∀𝑧∀𝑤∀𝑦𝜑)
3		alcom 1478	. 2 ⊢ (∀𝑥∀𝑧∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑥∀𝑤∀𝑦𝜑)
4		alcom 1478	. . 3 ⊢ (∀𝑥∀𝑤∀𝑦𝜑 ↔ ∀𝑤∀𝑥∀𝑦𝜑)
5	4	albii 1470	. 2 ⊢ (∀𝑧∀𝑥∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)
6	2, 3, 5	3bitri 206	1 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)

Syntax hints:   ↔ wb 105  ∀wal 1351

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

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  fun11  5285