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Theorem ss2ab 3238
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 2334 . . 3 𝑥{𝑥𝜑}
2 nfab1 2334 . . 3 𝑥{𝑥𝜓}
31, 2dfss2f 3161 . 2 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}))
4 abid 2177 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2177 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5imbi12i 239 . . 3 ((𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1481 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitri 184 1 ({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362  wcel 2160  {cab 2175  wss 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-in 3150  df-ss 3157
This theorem is referenced by:  abss  3239  ssab  3240  ss2abi  3242  ss2abdv  3243  ss2rab  3246  rabss2  3253  iotanul  5211  iotass  5213
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