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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralcom4f | Structured version Visualization version GIF version |
Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.) |
Ref | Expression |
---|---|
ralcom4f.1 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
ralcom4f | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom4f.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑥V | |
3 | 1, 2 | ralcomf 3077 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑) |
4 | ralv 3192 | . . 3 ⊢ (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑) | |
5 | 4 | ralbii 2963 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦𝜑) |
6 | ralv 3192 | . 2 ⊢ (∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | |
7 | 3, 5, 6 | 3bitr3i 289 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 Ⅎwnfc 2738 ∀wral 2896 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 |
This theorem is referenced by: (None) |
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