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Theorem ss2rab 4081
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 3434 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 df-rab 3434 . . 3 {𝑥𝐴𝜓} = {𝑥 ∣ (𝑥𝐴𝜓)}
31, 2sseq12i 4026 . 2 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐴𝜓)})
4 ss2ab 4072 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥 ∣ (𝑥𝐴𝜓)} ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 df-ral 3060 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
6 imdistan 567 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
76albii 1816 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
85, 7bitr2i 276 . 2 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) ↔ ∀𝑥𝐴 (𝜑𝜓))
93, 4, 83bitri 297 1 ({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wcel 2106  {cab 2712  wral 3059  {crab 3433  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-ss 3980
This theorem is referenced by:  ss2rabdv  4086  ss2rabi  4087  ondomon  10601  eltsms  24157  xrlimcnp  27026  chpssati  32392  lpssat  38995  lssatle  38997  lssat  38998  atlatle  39302  pmaple  39744  diaord  41030  mapdordlem2  41620  rmxyelqirrOLD  42899  ss2rabdf  45093  pimiooltgt  46666  preimageiingt  46676  preimaleiinlt  46677
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