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Mirrors > Home > ILE Home > Th. List > ss2abdv | GIF version |
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.) |
Ref | Expression |
---|---|
ss2abdv.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ss2abdv | ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2abdv.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | alrimiv 1854 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
3 | ss2ab 3196 | . 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒} ↔ ∀𝑥(𝜓 → 𝜒)) | |
4 | 2, 3 | sylibr 133 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 {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: ssopab2 4235 iotass 5152 imadif 5250 imain 5252 opabbrex 5865 ssoprab2 5877 tfr1onlemssrecs 6286 tfrcllemssrecs 6299 ss2ixp 6656 |
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