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Mirrors > Home > ILE Home > Th. List > ssralv | GIF version |
Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.) |
Ref | Expression |
---|---|
ssralv | ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3061 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | imim1d 75 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑))) |
3 | 2 | ralimdv2 2479 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ∀wral 2393 ⊆ wss 3041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-in 3047 df-ss 3054 |
This theorem is referenced by: iinss1 3795 poss 4190 sess2 4230 trssord 4272 funco 5133 funimaexglem 5176 isores3 5684 isoini2 5688 smores 6157 smores2 6159 tfrlem5 6179 resixp 6595 ac6sfi 6760 difinfinf 6954 peano5nnnn 7668 peano5nni 8691 caucvgre 10721 rexanuz 10728 cau3lem 10854 isumclim3 11160 fsumiun 11214 ctinf 11870 strsetsid 11919 tgcn 12304 tgcnp 12305 cnss2 12323 cncnp 12326 sslm 12343 metrest 12602 rescncf 12664 suplociccex 12699 limcresi 12731 nninfsellemeq 13137 |
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