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Theorem ss2rabdv 3178
Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
Hypothesis
Ref Expression
ss2rabdv.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ss2rabdv (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem ss2rabdv
StepHypRef Expression
1 ss2rabdv.1 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21ralrimiva 2505 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 ss2rab 3173 . 2 ({𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒} ↔ ∀𝑥𝐴 (𝜓𝜒))
42, 3sylibr 133 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1480  wral 2416  {crab 2420  wss 3071
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-in 3077  df-ss 3084
This theorem is referenced by:  sess1  4259  suppssfv  5978  suppssov1  5979  clsss  12301  metss2lem  12680
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