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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabssd | Structured version Visualization version GIF version |
Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
rabssd.1 | ⊢ Ⅎ𝑥𝜑 |
rabssd.2 | ⊢ Ⅎ𝑥𝐵 |
rabssd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝑥 ∈ 𝐵) |
Ref | Expression |
---|---|
rabssd | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabssd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rabssd.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝑥 ∈ 𝐵) | |
3 | 2 | 3exp 1264 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝑥 ∈ 𝐵))) |
4 | 1, 3 | ralrimi 2957 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝑥 ∈ 𝐵)) |
5 | rabssd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | rabssf 39302 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜒 → 𝑥 ∈ 𝐵)) |
7 | 4, 6 | sylibr 224 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 {crab 2916 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-in 3581 df-ss 3588 |
This theorem is referenced by: (None) |
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