Theorem ralrot3 3102

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,𝑦,𝐶   𝑥,𝑧,𝑦

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 3098	. . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑)
2	1	ralbii 2980	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑)
3		ralcom 3098	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 264	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Syntax hints:   ↔ wb 196  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by:  rmodislmodlem  18930  rmodislmod  18931