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Mirrors > Home > MPE Home > Th. List > ralrimdvv | Structured version Visualization version GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.) |
Ref | Expression |
---|---|
ralrimdvv.1 | ⊢ (𝜑 → (𝜓 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒))) |
Ref | Expression |
---|---|
ralrimdvv | ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrimdvv.1 | . . . 4 ⊢ (𝜑 → (𝜓 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒))) | |
2 | 1 | imp 407 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒)) |
3 | 2 | ralrimivv 3116 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒) |
4 | 3 | ex 413 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2110 ∀wral 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ral 3071 |
This theorem is referenced by: ralrimdvva 3120 lspsneu 20383 pmatcoe1fsupp 21848 aalioulem4 25493 fargshiftf1 44862 |
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