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Mirrors > Home > MPE Home > Th. List > r2al | Structured version Visualization version GIF version |
Description: Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 9-Jan-2020.) |
Ref | Expression |
---|---|
r2al | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21v 1940 | . 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) | |
2 | 1 | r2allem 3200 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∈ wcel 2114 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-ral 3143 |
This theorem is referenced by: r3al 3202 r2ex 3303 ralcom 3354 raliunxp 5710 codir 5980 qfto 5981 fununi 6429 dff13 7013 mpo2eqb 7283 tz7.48lem 8077 qliftfun 8382 zorn2lem4 9921 isirred2 19451 cnmpt12 22275 cnmpt22 22282 dchrelbas3 25814 cvmlift2lem12 32561 dfso2 32990 dfpo2 32991 inxpss 35584 inxpss3 35586 isdomn3 39824 |
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