Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ss2ab Structured version   GIF version

Theorem ss2ab 3002
 Description: Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
Assertion
Ref Expression
ss2ab ({xφ} ⊆ {xψ} ↔ x(φψ))

Proof of Theorem ss2ab
StepHypRef Expression
1 nfab1 2177 . . 3 x{xφ}
2 nfab1 2177 . . 3 x{xψ}
31, 2dfss2f 2930 . 2 ({xφ} ⊆ {xψ} ↔ x(x {xφ} → x {xψ}))
4 abid 2025 . . . 4 (x {xφ} ↔ φ)
5 abid 2025 . . . 4 (x {xψ} ↔ ψ)
64, 5imbi12i 228 . . 3 ((x {xφ} → x {xψ}) ↔ (φψ))
76albii 1356 . 2 (x(x {xφ} → x {xψ}) ↔ x(φψ))
83, 7bitri 173 1 ({xφ} ⊆ {xψ} ↔ x(φψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  {cab 2023   ⊆ 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-in 2918  df-ss 2925 This theorem is referenced by:  abss  3003  ssab  3004  ss2abi  3006  ss2abdv  3007  ss2rab  3010  rabss2  3017  iotanul  4825  iotass  4827
 Copyright terms: Public domain W3C validator