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Theorem ss2rabdv 3015
Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
Hypothesis
Ref Expression
ss2rabdv.1 ((φ x A) → (ψχ))
Assertion
Ref Expression
ss2rabdv (φ → {x Aψ} ⊆ {x Aχ})
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   χ(x)   A(x)

Proof of Theorem ss2rabdv
StepHypRef Expression
1 ss2rabdv.1 . . 3 ((φ x A) → (ψχ))
21ralrimiva 2386 . 2 (φx A (ψχ))
3 ss2rab 3010 . 2 ({x Aψ} ⊆ {x Aχ} ↔ x A (ψχ))
42, 3sylibr 137 1 (φ → {x Aψ} ⊆ {x Aχ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wral 2300  {crab 2304  wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-in 2918  df-ss 2925
This theorem is referenced by:  sess1  4059  suppssfv  5650  suppssov1  5651
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