Theorem ralrot3 3295

Description: Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)

Assertion

Ref	Expression
ralrot3	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑥,𝐶   𝑦,𝐶   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)   𝐶(𝑧)

Proof of Theorem ralrot3

Step	Hyp	Ref	Expression
1		ralcom 3292	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑)
2	1	ralbii 3110	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑)
3		ralcom 3292	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-11 2191

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-ral 3079

This theorem is used by:  ralcom13  3296  isdomn4r  20828  rmodislmodlem  21061  rmodislmod  21062  isclmp  25267  addsprop  28180  negsprop  28239  mulsprop  28334  ntrneikb  44848  ntrneixb  44849