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Mirrors > Home > MPE Home > Th. List > ralim | Structured version Visualization version GIF version |
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.) (Proof shortened by Wolf Lammen, 1-Dec-2019.) |
Ref | Expression |
---|---|
ralim | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | ral2imi 3086 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wral 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-ral 3063 |
This theorem is referenced by: r19.30OLD 3122 ralimdaa 3258 mpteqb 7018 tz7.49 8445 mptelixpg 8929 resixpfo 8930 bnd 9887 kmlem12 10156 lbzbi 12920 r19.29uz 15297 caubnd 15305 alzdvds 16263 ptclsg 23119 isucn2 23784 fusgreghash2wsp 29622 omssubadd 33330 subgrwlk 34154 dfon2lem8 34793 fvineqsneq 36341 dford3lem2 41814 neik0pk1imk0 42846 grur1cld 43039 mnuprdlem4 43082 mnurndlem1 43088 |
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