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Theorem ss2rab 3071
Description: Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
Assertion
Ref Expression
ss2rab ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))

Proof of Theorem ss2rab
StepHypRef Expression
1 df-rab 2358 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 2358 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2sseq12i 3026 . 2 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐴𝜓)})
4 ss2ab 3063 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐴𝜓)} ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 df-ral 2354 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
6 imdistan 433 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
76albii 1400 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
85, 7bitr2i 183 . 2 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) ↔ ∀𝑥𝐴 (𝜑𝜓))
93, 4, 83bitri 204 1 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wcel 1434  {cab 2068  wral 2349  {crab 2353  wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-in 2980  df-ss 2987
This theorem is referenced by:  ss2rabdv  3076  ss2rabi  3077
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