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Mirrors > Home > MPE Home > Th. List > 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 3140 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜓 | |
3 | 1, 2 | nfan 1907 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) |
4 | id 22 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝐴)) | |
5 | 4 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ∧ 𝑥 ∈ 𝐴)) |
6 | rspa 3128 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ 𝑥 ∈ 𝐴) → 𝜓) | |
7 | 6 | adantll 714 | . . . 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 3406 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 𝜒) |
11 | 10 | ex 416 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 Ⅎwnf 1791 ∈ wcel 2110 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-ral 3066 |
This theorem is referenced by: xlimmnfvlem1 43048 xlimmnfvlem2 43049 xlimpnfvlem1 43052 xlimpnfvlem2 43053 |
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