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Theorem rabssdv 3227
Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)
Hypothesis
Ref Expression
rabssdv.1 ((𝜑𝑥𝐴𝜓) → 𝑥𝐵)
Assertion
Ref Expression
rabssdv (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rabssdv
StepHypRef Expression
1 rabssdv.1 . . . 4 ((𝜑𝑥𝐴𝜓) → 𝑥𝐵)
213exp 1197 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝑥𝐵)))
32ralrimiv 2542 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝑥𝐵))
4 rabss 3224 . 2 ({𝑥𝐴𝜓} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜓𝑥𝐵))
53, 4sylibr 133 1 (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 973  wcel 2141  wral 2448  {crab 2452  wss 3121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-in 3127  df-ss 3134
This theorem is referenced by:  zsupssdc  11898
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