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Mirrors > Home > ILE Home > Th. List > ralrimivv | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.) |
Ref | Expression |
---|---|
ralrimivv.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜓)) |
Ref | Expression |
---|---|
ralrimivv | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrimivv.1 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜓)) | |
2 | 1 | expd 258 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜓))) |
3 | 2 | ralrimdv 2556 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜓)) |
4 | 3 | ralrimiv 2549 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-ral 2460 |
This theorem is referenced by: ralrimivva 2559 ralrimdvv 2561 reuind 2942 ssrel2 4716 f1o2ndf1 6228 smoiso 6302 nndifsnid 6507 receuap 8625 lbreu 8901 0subm 12870 insubm 12871 iscmnd 13099 tgcl 13534 topbas 13537 epttop 13560 restbasg 13638 txbas 13728 txbasval 13737 blfps 13879 blf 13880 blbas 13903 |
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