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Mirrors > Home > MPE Home > Th. List > r2alf | Structured version Visualization version GIF version |
Description: Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.) Use r2allem 2937. (Revised by Wolf Lammen, 9-Jan-2020.) |
Ref | Expression |
---|---|
r2alf.1 | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
r2alf | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r2alf.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
2 | 1 | nfcri 2758 | . . 3 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
3 | 2 | 19.21 2075 | . 2 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))) |
4 | 3 | 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 Ⅎwnfc 2751 ∀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 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 |
This theorem is referenced by: r2exf 3060 ralcomf 3096 |
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