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Mirrors > Home > MPE Home > Th. List > ralimdaa | Structured version Visualization version GIF version |
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.) |
Ref | Expression |
---|---|
ralimdaa.1 | ⊢ Ⅎ𝑥𝜑 |
ralimdaa.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ralimdaa | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimdaa.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ralimdaa.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) | |
3 | 2 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
4 | 1, 3 | ralrimi 3255 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
5 | ralim 3084 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1780 ∈ 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 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-ral 3060 |
This theorem is referenced by: ralbida 3268 eltsk2g 10789 ptcnplem 23645 poimirlem26 37633 allbutfifvre 45631 climleltrp 45632 fnlimabslt 45635 limsupub2 45768 liminflbuz2 45771 xlimmnfvlem1 45788 xlimmnfvlem2 45789 xlimpnfvlem1 45792 xlimpnfvlem2 45793 stoweidlem61 46017 stoweid 46019 fourierdlem73 46135 smflimlem2 46728 |
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