Theorem alrot4 2160

Description: Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)

Assertion

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

Proof of Theorem alrot4

Step	Hyp	Ref	Expression
1		alrot3 2159	. . 3 ⊢ (∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑦𝜑)
2	1	albii 1818	. 2 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑥∀𝑧∀𝑤∀𝑦𝜑)
3		alrot3 2159	. 2 ⊢ (∀𝑥∀𝑧∀𝑤∀𝑦𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)
4	2, 3	bitri 275	1 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-11 2156

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  2mo  2647  fun11  6639