Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralimda | Structured version Visualization version GIF version |
Description: Deduction quantifying both antecedent and consequent. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
ralimda.1 | ⊢ Ⅎ𝑥𝜑 |
ralimda.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ralimda | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimda.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | nfra1 3216 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜓 | |
3 | 1, 2 | nfan 1891 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) |
4 | id 22 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝐴)) | |
5 | 4 | adantlr 711 | . . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝐴)) |
6 | rspa 3203 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
7 | 6 | adantll 710 | . . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) → 𝜓) |
8 | ralimda.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) | |
9 | 5, 7, 8 | sylc 65 | . . 3 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) → 𝜒) |
10 | 3, 9 | ralrimia 41274 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 𝜒) |
11 | 10 | ex 413 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1775 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-ral 3140 |
This theorem is referenced by: xlimmnfvlem1 41989 xlimmnfvlem2 41990 xlimpnfvlem1 41993 xlimpnfvlem2 41994 |
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