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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 3165 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∈ wcel 2111 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-ral 3111 |
This theorem is referenced by: r3al 3167 r2ex 3262 ralcom 3307 raliunxp 5674 codir 5947 qfto 5948 fununi 6399 dff13 6991 mpo2eqb 7262 tz7.48lem 8060 qliftfun 8365 zorn2lem4 9910 isirred2 19447 cnmpt12 22272 cnmpt22 22279 dchrelbas3 25822 cvmlift2lem12 32674 dfso2 33103 dfpo2 33104 inxpss 35729 inxpss3 35731 isdomn3 40148 |
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