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Theorem ssabdv 41572
Description: Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.)
Hypothesis
Ref Expression
ssabdv.1 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
ssabdv (𝜑𝐴 ⊆ {𝑥𝜓})
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem ssabdv
StepHypRef Expression
1 abid1 2862 . 2 𝐴 = {𝑥𝑥𝐴}
2 ssabdv.1 . . 3 (𝜑 → (𝑥𝐴𝜓))
32ss2abdv 4053 . 2 (𝜑 → {𝑥𝑥𝐴} ⊆ {𝑥𝜓})
41, 3eqsstrid 4023 1 (𝜑𝐴 ⊆ {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  {cab 2701  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-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958
This theorem is referenced by: (None)
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