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Theorem ssabf 44466
Description: Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
ssabf.1 𝑥𝐴
Assertion
Ref Expression
ssabf (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Proof of Theorem ssabf
StepHypRef Expression
1 ssabf.1 . . . 4 𝑥𝐴
21abid2f 2933 . . 3 {𝑥𝑥𝐴} = 𝐴
32sseq1i 4008 . 2 ({𝑥𝑥𝐴} ⊆ {𝑥𝜑} ↔ 𝐴 ⊆ {𝑥𝜑})
4 ss2ab 4054 . 2 ({𝑥𝑥𝐴} ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
53, 4bitr3i 277 1 (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532  wcel 2099  {cab 2705  wnfc 2879  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-v 3473  df-in 3954  df-ss 3964
This theorem is referenced by:  ssrabf  44480
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