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Theorem ss2rabdf 43355
Description: Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Hypotheses
Ref Expression
ss2rabdf.1 𝑥𝜑
ss2rabdf.2 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ss2rabdf (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})

Proof of Theorem ss2rabdf
StepHypRef Expression
1 ss2rabdf.1 . . 3 𝑥𝜑
2 ss2rabdf.2 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
32ex 413 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3240 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ss2rab 4028 . 2 ({𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒} ↔ ∀𝑥𝐴 (𝜓𝜒))
64, 5sylibr 233 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wnf 1785  wcel 2106  wral 3064  {crab 3407  wss 3910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rab 3408  df-v 3447  df-in 3917  df-ss 3927
This theorem is referenced by:  pimxrneun  43714
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