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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 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
4 | 1, 3 | ralrimi 3258 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
5 | ss2rab 4088 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒} ↔ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) | |
6 | 4, 5 | sylibr 234 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1781 ∈ wcel 2103 ∀wral 3063 {crab 3438 ⊆ wss 3970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rab 3439 df-ss 3987 |
This theorem is referenced by: pimxrneun 45339 |
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