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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 2939 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | imim1d 69 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑))) |
3 | 2 | ralimdv2 2389 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ∀wral 2306 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-ral 2311 df-in 2924 df-ss 2931 |
This theorem is referenced by: iinss1 3669 poss 4035 sess2 4075 trssord 4117 funco 4940 funimaexglem 4982 isores3 5455 isoini2 5458 smores 5907 smores2 5909 tfrlem5 5930 ac6sfi 6352 peano5nnnn 6966 peano5nni 7917 caucvgre 9580 rexanuz 9587 cau3lem 9710 |
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