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Theorem ss2ab 4061
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 2906 . . 3 𝑥{𝑥𝜑}
2 nfab1 2906 . . 3 𝑥{𝑥𝜓}
31, 2dfssf 3973 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}))
4 abid 2717 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2717 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5imbi12i 350 . . 3 ((𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1818 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitri 275 1 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1537  wcel 2107  {cab 2713  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-10 2140  ax-11 2156  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-clel 2815  df-nfc 2891  df-ss 3967
This theorem is referenced by:  abss  4062  ssab  4063  ss2rab  4070  rabss2  4077  rabsssn  4667  rabsspr  32521  rabsstp  32522  bj-gabss  36937  clss2lem  43629  ssabf  45110  abssf  45122  cfsetssfset  47073  sprssspr  47473
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