MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abssdvOLD Structured version   Visualization version   GIF version

Theorem abssdvOLD 4058
Description: Obsolete version of abssdv 4057 as of 12-Dec-2024. (Contributed by NM, 20-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
abssdv.1 (𝜑 → (𝜓𝑥𝐴))
Assertion
Ref Expression
abssdvOLD (𝜑 → {𝑥𝜓} ⊆ 𝐴)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem abssdvOLD
StepHypRef Expression
1 abssdv.1 . . 3 (𝜑 → (𝜓𝑥𝐴))
21alrimiv 1922 . 2 (𝜑 → ∀𝑥(𝜓𝑥𝐴))
3 abss 4050 . 2 ({𝑥𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓𝑥𝐴))
42, 3sylibr 233 1 (𝜑 → {𝑥𝜓} ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wcel 2098  {cab 2702  wss 3939
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 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ss 3956
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator