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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 1868 | . 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) | |
2 | 1 | r2allem 2937 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-ral 2917 |
This theorem is referenced by: r3al 2940 r2ex 3061 soss 5053 raliunxp 5261 codir 5516 qfto 5517 fununi 5964 dff13 6512 mpt22eqb 6769 tz7.48lem 7536 qliftfun 7832 zorn2lem4 9321 isirred2 18701 cnmpt12 21470 cnmpt22 21477 dchrelbas3 24963 cvmlift2lem12 31296 dfso2 31644 dfpo2 31645 inxpss 34082 inxpss3 34084 isdomn3 37782 |
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