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Mirrors > Home > ILE Home > Th. List > ralrimivv | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.) |
Ref | Expression |
---|---|
ralrimivv.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜓)) |
Ref | Expression |
---|---|
ralrimivv | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrimivv.1 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜓)) | |
2 | 1 | expd 245 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜓))) |
3 | 2 | ralrimdv 2398 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜓)) |
4 | 3 | ralrimiv 2391 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ∀wral 2306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-17 1419 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-ral 2311 |
This theorem is referenced by: ralrimivva 2401 ralrimdvv 2403 reuind 2744 ssrel2 4430 f1o2ndf1 5849 smoiso 5917 receuap 7650 |
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