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Theorem ssab 4074
Description: Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssab (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssab
StepHypRef Expression
1 abid2 2877 . . 3 {𝑥𝑥𝐴} = 𝐴
21sseq1i 4024 . 2 ({𝑥𝑥𝐴} ⊆ {𝑥𝜑} ↔ 𝐴 ⊆ {𝑥𝜑})
3 ss2ab 4072 . 2 ({𝑥𝑥𝐴} ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3bitr3i 277 1 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wcel 2106  {cab 2712  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-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ss 3980
This theorem is referenced by:  ssabral  4075  ssrab  4083  wdomd  9619  ixpiunwdom  9628  lidldvgen  21362  prdsxmslem2  24558  ballotlem2  34470
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