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Mirrors > Home > ILE Home > Th. List > ss2ab | GIF version |
Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.) |
Ref | Expression |
---|---|
ss2ab | ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2284 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
2 | nfab1 2284 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
3 | 1, 2 | dfss2f 3093 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓})) |
4 | abid 2128 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
5 | abid 2128 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
6 | 4, 5 | imbi12i 238 | . . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 → 𝜓)) |
7 | 6 | albii 1447 | . 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} → 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 → 𝜓)) |
8 | 3, 7 | bitri 183 | 1 ⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1330 ∈ wcel 1481 {cab 2126 ⊆ wss 3076 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-in 3082 df-ss 3089 |
This theorem is referenced by: abss 3171 ssab 3172 ss2abi 3174 ss2abdv 3175 ss2rab 3178 rabss2 3185 iotanul 5111 iotass 5113 |
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