Theorem alrot4 1415

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 1414	. . 3 ⊢ (∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑦𝜑)
2	1	albii 1399	. 2 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑥∀𝑧∀𝑤∀𝑦𝜑)
3		alcom 1407	. 2 ⊢ (∀𝑥∀𝑧∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑥∀𝑤∀𝑦𝜑)
4		alcom 1407	. . 3 ⊢ (∀𝑥∀𝑤∀𝑦𝜑 ↔ ∀𝑤∀𝑥∀𝑦𝜑)
5	4	albii 1399	. 2 ⊢ (∀𝑧∀𝑥∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)
6	2, 3, 5	3bitri 204	1 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)

Syntax hints:   ↔ wb 103  ∀wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  fun11  4986