Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabssdv Structured version   GIF version

Theorem rabssdv 3014
 Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
Hypothesis
Ref Expression
rabssdv.1 ((φ x A ψ) → x B)
Assertion
Ref Expression
rabssdv (φ → {x Aψ} ⊆ 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 ((φ x A ψ) → x B)
213exp 1102 . . 3 (φ → (x A → (ψx B)))
32ralrimiv 2385 . 2 (φx A (ψx B))
4 rabss 3011 . 2 ({x Aψ} ⊆ Bx A (ψx B))
53, 4sylibr 137 1 (φ → {x Aψ} ⊆ B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 884   ∈ 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-3an 886  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: (None)
 Copyright terms: Public domain W3C validator