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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ss2rabdf | Structured version Visualization version GIF version |
Description: Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
Ref | Expression |
---|---|
ss2rabdf.1 | ⊢ Ⅎ𝑥𝜑 |
ss2rabdf.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ss2rabdf | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2rabdf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ss2rabdf.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) | |
3 | 2 | ex 411 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
4 | 1, 3 | ralrimi 3252 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
5 | ss2rab 4068 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3058 {crab 3430 ⊆ wss 3949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rab 3431 df-v 3475 df-in 3956 df-ss 3966 |
This theorem is referenced by: pimxrneun 44918 |
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