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Theorem ssrabdf 44360
Description: Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Hypotheses
Ref Expression
ssrabdf.1 𝑥𝐴
ssrabdf.2 𝑥𝐵
ssrabdf.3 𝑥𝜑
ssrabdf.4 (𝜑𝐵𝐴)
ssrabdf.5 ((𝜑𝑥𝐵) → 𝜓)
Assertion
Ref Expression
ssrabdf (𝜑𝐵 ⊆ {𝑥𝐴𝜓})

Proof of Theorem ssrabdf
StepHypRef Expression
1 ssrabdf.4 . 2 (𝜑𝐵𝐴)
2 ssrabdf.3 . . 3 𝑥𝜑
3 ssrabdf.5 . . 3 ((𝜑𝑥𝐵) → 𝜓)
42, 3ralrimia 3249 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
5 ssrabdf.2 . . 3 𝑥𝐵
6 ssrabdf.1 . . 3 𝑥𝐴
75, 6ssrabf 44359 . 2 (𝐵 ⊆ {𝑥𝐴𝜓} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜓))
81, 4, 7sylanbrc 582 1 (𝜑𝐵 ⊆ {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1777  wcel 2098  wnfc 2877  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:  smfpimne2  46109
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