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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 2214 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) |
5 | df-ral 3146 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
6 | 4, 5 | syl6ibr 254 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 Ⅎwnf 1783 ∈ wcel 2113 ∀wral 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-ex 1780 df-nf 1784 df-ral 3146 |
This theorem is referenced by: reusv2lem3 5304 fliftfun 7068 mapxpen 8686 domtriomlem 9867 dedekind 10806 fzrevral 12995 matunitlindflem2 34893 riotasv3d 36100 ssralv2 40871 setrec1lem2 44798 |
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