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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 3165 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) | |
2 | ss2abi.1 | . 2 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | mpgbir 1429 | 1 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 {cab 2125 ⊆ wss 3071 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-in 3077 df-ss 3084 |
This theorem is referenced by: abssi 3172 rabssab 3184 pwsnss 3730 iinuniss 3895 pwpwssunieq 3901 abssexg 4106 imassrn 4892 imadiflem 5202 imainlem 5204 fabexg 5310 f1oabexg 5379 tfrcllemssrecs 6249 mapex 6548 tgval 12218 tgvalex 12219 |
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