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Mirrors > Home > ILE Home > Th. List > ss2abi | GIF version |
Description: Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.) |
Ref | Expression |
---|---|
ss2abi.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
ss2abi | ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2ab 3196 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) | |
2 | ss2abi.1 | . 2 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | mpgbir 1433 | 1 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 {cab 2143 ⊆ wss 3102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-in 3108 df-ss 3115 |
This theorem is referenced by: abssi 3203 rabssab 3216 pwsnss 3768 iinuniss 3933 pwpwssunieq 3939 abssexg 4145 imassrn 4941 imadiflem 5251 imainlem 5253 fabexg 5359 f1oabexg 5428 tfrcllemssrecs 6301 mapex 6601 tgval 12519 tgvalex 12520 |
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