Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abssf Structured version   Visualization version   GIF version

Theorem abssf 44314
Description: Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
abssf.1 𝑥𝐴
Assertion
Ref Expression
abssf ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Proof of Theorem abssf
StepHypRef Expression
1 abssf.1 . . . 4 𝑥𝐴
21abid2f 2928 . . 3 {𝑥𝑥𝐴} = 𝐴
32sseq2i 4004 . 2 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ {𝑥𝜑} ⊆ 𝐴)
4 ss2ab 4049 . 2 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
53, 4bitr3i 277 1 ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wcel 2098  {cab 2701  wnfc 2875  wss 3941
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-v 3468  df-in 3948  df-ss 3958
This theorem is referenced by:  rabssf  44321
  Copyright terms: Public domain W3C validator