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Theorem rabssrabd 4042
Description: Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022.)
Hypotheses
Ref Expression
rabssrabd.1 (𝜑𝐴𝐵)
rabssrabd.2 ((𝜑𝜓𝑥𝐴) → 𝜒)
Assertion
Ref Expression
rabssrabd (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rabssrabd
StepHypRef Expression
1 3anan32 1098 . . . . 5 ((𝜑𝜓𝑥𝐴) ↔ ((𝜑𝑥𝐴) ∧ 𝜓))
2 rabssrabd.2 . . . . 5 ((𝜑𝜓𝑥𝐴) → 𝜒)
31, 2sylbir 234 . . . 4 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43ex 414 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
54ss2rabdv 4034 . 2 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
6 rabssrabd.1 . . 3 (𝜑𝐴𝐵)
7 rabss2 4036 . . 3 (𝐴𝐵 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
86, 7syl 17 . 2 (𝜑 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
95, 8sstrd 3955 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  {crab 3406  wss 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rab 3407  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by:  suppfnss  8121  clwlknon2num  29354  numclwlk1lem2  29356
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