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Theorem rabssrabd 4083
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 1097 . . . . 5 ((𝜑𝜓𝑥𝐴) ↔ ((𝜑𝑥𝐴) ∧ 𝜓))
2 rabssrabd.2 . . . . 5 ((𝜑𝜓𝑥𝐴) → 𝜒)
31, 2sylbir 235 . . . 4 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43ex 412 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
54ss2rabdv 4076 . 2 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
6 rabssrabd.1 . . 3 (𝜑𝐴𝐵)
7 rabss2 4078 . . 3 (𝐴𝐵 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
86, 7syl 17 . 2 (𝜑 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
95, 8sstrd 3994 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wcel 2108  {crab 3436  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-ss 3968
This theorem is referenced by:  suppfnss  8214  clwlknon2num  30387  numclwlk1lem2  30389
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