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Mirrors > Home > ILE Home > Th. List > ss2rabi | GIF version |
Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.) |
Ref | Expression |
---|---|
ss2rabi.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
ss2rabi | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2rab 3097 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) | |
2 | ss2rabi.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
3 | 1, 2 | mprgbir 2433 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1438 {crab 2363 ⊆ wss 2999 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rab 2368 df-in 3005 df-ss 3012 |
This theorem is referenced by: supubti 6694 suplubti 6695 |
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