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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 2977 | . . 3 ⊢ Ⅎ𝑥V | |
3 | 1, 2 | ralcomf 3357 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑) |
4 | ralv 3519 | . . 3 ⊢ (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑) | |
5 | 4 | ralbii 3165 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦𝜑) |
6 | ralv 3519 | . 2 ⊢ (∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) | |
7 | 3, 5, 6 | 3bitr3i 303 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1531 Ⅎwnfc 2961 ∀wral 3138 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 |
This theorem is referenced by: (None) |
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