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Mirrors > Home > MPE Home > Th. List > ralrot3 | Structured version Visualization version GIF version |
Description: Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.) |
Ref | Expression |
---|---|
ralrot3 | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom 3271 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑) | |
2 | 1 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 𝜑) |
3 | ralcom 3271 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wral 3061 |
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 1914 ax-11 2155 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-ral 3062 |
This theorem is referenced by: ralcom13 3276 rmodislmodlem 20404 rmodislmod 20405 rmodislmodOLD 20406 isclmp 24476 addsprop 27310 negsprop 27355 ntrneikb 42454 ntrneixb 42455 |
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