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Theorem rabssrabd 4076
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 1094 . . . . 5 ((𝜑𝜓𝑥𝐴) ↔ ((𝜑𝑥𝐴) ∧ 𝜓))
2 rabssrabd.2 . . . . 5 ((𝜑𝜓𝑥𝐴) → 𝜒)
31, 2sylbir 234 . . . 4 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43ex 412 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
54ss2rabdv 4068 . 2 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
6 rabssrabd.1 . . 3 (𝜑𝐴𝐵)
7 rabss2 4070 . . 3 (𝐴𝐵 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
86, 7syl 17 . 2 (𝜑 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
95, 8sstrd 3987 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wcel 2098  {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-3an 1086  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:  suppfnss  8174  clwlknon2num  30130  numclwlk1lem2  30132
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