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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 3215 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) | |
2 | ss2abi.1 | . 2 ⊢ (𝜑 → 𝜓) | |
3 | 1, 2 | mpgbir 1446 | 1 ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 {cab 2156 ⊆ wss 3121 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-in 3127 df-ss 3134 |
This theorem is referenced by: abssi 3222 rabssab 3235 pwsnss 3790 iinuniss 3955 pwpwssunieq 3961 abssexg 4168 imassrn 4964 imadiflem 5277 imainlem 5279 fabexg 5385 f1oabexg 5454 tfrcllemssrecs 6331 mapex 6632 tgval 12843 tgvalex 12844 |
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