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Theorem ss2rabdf 44416
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 412 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
41, 3ralrimi 3248 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ss2rab 4063 . 2 ({𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒} ↔ ∀𝑥𝐴 (𝜓𝜒))
64, 5sylibr 233 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1777  wcel 2098  wral 3055  {crab 3426  wss 3943
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 2697
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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960
This theorem is referenced by:  pimxrneun  44768
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