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Theorem ssab 4039
 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 2955 . . 3 {𝑥𝑥𝐴} = 𝐴
21sseq1i 3993 . 2 ({𝑥𝑥𝐴} ⊆ {𝑥𝜑} ↔ 𝐴 ⊆ {𝑥𝜑})
3 ss2ab 4037 . 2 ({𝑥𝑥𝐴} ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
42, 3bitr3i 279 1 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1528   ∈ wcel 2107  {cab 2797   ⊆ wss 3934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-in 3941  df-ss 3950 This theorem is referenced by:  ssabral  4040  ssrab  4047  wdomd  9037  ixpiunwdom  9047  lidldvgen  20020  prdsxmslem2  23131  ballotlem2  31734
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