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| Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) For a version based on fewer axioms see ralrimdv 3151. (Contributed by NM, 16-Feb-2004.) | 
| Ref | Expression | 
|---|---|
| ralrimd.1 | ⊢ Ⅎ𝑥𝜑 | 
| ralrimd.2 | ⊢ Ⅎ𝑥𝜓 | 
| ralrimd.3 | ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒))) | 
| Ref | Expression | 
|---|---|
| ralrimd | ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ralrimd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | ralrimd.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
| 3 | ralrimd.3 | . . 3 ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒))) | |
| 4 | 1, 2, 3 | alrimd 2214 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) | 
| 5 | df-ral 3061 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
| 6 | 4, 5 | imbitrrdi 252 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1782 ∈ wcel 2107 ∀wral 3060 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 df-nf 1783 df-ral 3061 | 
| This theorem is referenced by: reusv2lem3 5399 fliftfun 7333 mapxpen 9184 domtriomlem 10483 dedekind 11425 fzrevral 13653 matunitlindflem2 37625 riotasv3d 38962 ssralv2 44556 setrec1lem2 49262 | 
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