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Theorem abssdvOLD 4027
Description: Obsolete version of abssdv 4026 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 1931 . 2 (𝜑 → ∀𝑥(𝜓𝑥𝐴))
3 abss 4018 . 2 ({𝑥𝜓} ⊆ 𝐴 ↔ ∀𝑥(𝜓𝑥𝐴))
42, 3sylibr 233 1 (𝜑 → {𝑥𝜓} ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wcel 2107  {cab 2710  wss 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by: (None)
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