Theorem alrot4 1739

Description: Rotate 4 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 1738	. . 3 ⊢ (∀y∀z∀wφ ↔ ∀z∀w∀yφ)
2	1	albii 1566	. 2 ⊢ (∀x∀y∀z∀wφ ↔ ∀x∀z∀w∀yφ)
3		alrot3 1738	. 2 ⊢ (∀x∀z∀w∀yφ ↔ ∀z∀w∀x∀yφ)
4	2, 3	bitri 240	1 ⊢ (∀x∀y∀z∀wφ ↔ ∀z∀w∀x∀yφ)

This proof depends on syntax axioms:   ↔ wb 176  ∀wal 1540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-7 1734

This proof depends on definitions:  df-bi 177

This theorem is used by:  2mo  2282  fun11  5160