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Mirrors > Home > MPE Home > Th. List > ralimdvva | Structured version Visualization version GIF version |
Description: Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1807). (Contributed by AV, 27-Nov-2019.) |
Ref | Expression |
---|---|
ralimdvva.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ralimdvva | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimdvva.1 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) | |
2 | 1 | anassrs 467 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)) |
3 | 2 | ralimdva 3165 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 𝜓 → ∀𝑦 ∈ 𝐵 𝜒)) |
4 | 3 | ralimdva 3165 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ral 3060 |
This theorem is referenced by: ralimdvv 3206 dedekindle 11423 isdomn4 20733 islmhm2 21055 dflidl2rng 21246 dmatscmcl 22525 cpmatacl 22738 cpmatinvcl 22739 mat2pmatf1 22751 pmatcollpw2lem 22799 tgpt0 24143 isngp4 24641 addcnlem 24900 c1lip3 26053 aalioulem2 26390 aalioulem5 26393 aalioulem6 26394 aaliou 26395 iscgrglt 28537 2pthfrgrrn 30311 2pthfrgrrn2 30312 equivbnd 37777 ghomco 37878 fcoresf1 47019 fullthinc 48846 |
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