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Mirrors > Home > MPE Home > Th. List > ralrimd | Structured version Visualization version GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (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 2071 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) |
5 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
6 | 4, 5 | syl6ibr 241 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 df-ral 2901 |
This theorem is referenced by: reusv2lem3 4797 fliftfun 6462 mapxpen 8011 domtriomlem 9147 dedekind 10079 fzrevral 12294 matunitlindflem2 32576 riotasv3d 33264 ssralv2 37758 setrec1lem2 42234 |
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