Theorem ralrot3 3269

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 3266	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑)
2	1	ralbii 3084	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑)
3		ralcom 3266	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 206  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-11 2163

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-ral 3053

This theorem is referenced by:  ralcom13  3270  isdomn4r  20664  rmodislmodlem  20892  rmodislmod  20893  isclmp  25065  addsprop  27984  negsprop  28043  mulsprop  28138  ntrneikb  44439  ntrneixb  44440