Theorem ralcom13 3286

Description: Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.)

Assertion

Ref	Expression
ralcom13	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)

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

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

Proof of Theorem ralcom13

Step	Hyp	Ref	Expression
1		ralcom 3166	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 𝜑)
2		ralcom 3166	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝜑)
3	2	ralbii 3092	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝜑)
4		ralcom 3166	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
5	1, 3, 4	3bitri 297	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 205  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-11 2154

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-ral 3069

This theorem is referenced by: (None)