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Theorem ss2rabdf 45057
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 3263 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
5 ss2rab 4094 . 2 ({𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒} ↔ ∀𝑥𝐴 (𝜓𝜒))
64, 5sylibr 234 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1781  wcel 2108  wral 3067  {crab 3443  wss 3976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-ss 3993
This theorem is referenced by:  pimxrneun  45406
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