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Theorem rabss2 3017
Description: Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
rabss2 (AB → {x Aφ} ⊆ {x Bφ})
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem rabss2
StepHypRef Expression
1 pm3.45 529 . . . 4 ((x Ax B) → ((x A φ) → (x B φ)))
21alimi 1341 . . 3 (x(x Ax B) → x((x A φ) → (x B φ)))
3 dfss2 2928 . . 3 (ABx(x Ax B))
4 ss2ab 3002 . . 3 ({x ∣ (x A φ)} ⊆ {x ∣ (x B φ)} ↔ x((x A φ) → (x B φ)))
52, 3, 43imtr4i 190 . 2 (AB → {x ∣ (x A φ)} ⊆ {x ∣ (x B φ)})
6 df-rab 2309 . 2 {x Aφ} = {x ∣ (x A φ)}
7 df-rab 2309 . 2 {x Bφ} = {x ∣ (x B φ)}
85, 6, 73sstr4g 2980 1 (AB → {x Aφ} ⊆ {x Bφ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   wcel 1390  {cab 2023  {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-bnd 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-rab 2309  df-in 2918  df-ss 2925
This theorem is referenced by:  sess2  4060
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