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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 1119 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝑥 ∈ 𝐵))) |
4 | 1, 3 | ralrimi 3237 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝑥 ∈ 𝐵)) |
5 | rabssd.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
6 | 5 | rabssf 42706 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜒 → 𝑥 ∈ 𝐵)) |
7 | 4, 6 | sylibr 233 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 Ⅎwnf 1783 ∈ wcel 2104 Ⅎwnfc 2885 ∀wral 3062 {crab 3284 ⊆ wss 3892 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rab 3287 df-v 3439 df-in 3899 df-ss 3909 |
This theorem is referenced by: pimxrneun 43077 |
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