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Theorem ss2ab 3633
Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
Assertion
Ref Expression
ss2ab ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))

Proof of Theorem ss2ab
StepHypRef Expression
1 nfab1 2753 . . 3 𝑥{𝑥𝜑}
2 nfab1 2753 . . 3 𝑥{𝑥𝜓}
31, 2dfss2f 3559 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}))
4 abid 2598 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2598 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5imbi12i 339 . . 3 ((𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1737 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitri 263 1 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  wcel 1977  {cab 2596  wss 3540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554
This theorem is referenced by:  abss  3634  ssab  3635  ss2abi  3637  ss2abdv  3638  ss2rab  3641  rabss2  3648  rabsssn  4162  clss2lem  36937  ssabf  38308  abssf  38326
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