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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 1867 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
3 | ss2ab 3215 | . 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 1346 {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: ssopab2 4260 iotass 5177 imadif 5278 imain 5280 opabbrex 5897 ssoprab2 5909 tfr1onlemssrecs 6318 tfrcllemssrecs 6331 ss2ixp 6689 |
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