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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 413 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
4 | 1, 3 | ralrimi 3240 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
5 | ss2rab 4028 | . 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 396 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3064 {crab 3407 ⊆ wss 3910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rab 3408 df-v 3447 df-in 3917 df-ss 3927 |
This theorem is referenced by: pimxrneun 43714 |
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