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Mirrors > Home > MPE Home > Th. List > raaanv | Structured version Visualization version GIF version |
Description: Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.) |
Ref | Expression |
---|---|
raaanv | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1995 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1995 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | raaan 4222 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-v 3353 df-dif 3727 df-nul 4065 |
This theorem is referenced by: reusv3i 5005 f1mpt 6662 isclo2 21114 ntrk2imkb 38862 |
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