Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfss2f Structured version   GIF version

Theorem dfss2f 2930
 Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
dfss2f.1 xA
dfss2f.2 xB
Assertion
Ref Expression
dfss2f (ABx(x Ax B))

Proof of Theorem dfss2f
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 dfss2 2928 . 2 (ABz(z Az B))
2 dfss2f.1 . . . . 5 xA
32nfcri 2169 . . . 4 x z A
4 dfss2f.2 . . . . 5 xB
54nfcri 2169 . . . 4 x z B
63, 5nfim 1461 . . 3 x(z Az B)
7 nfv 1418 . . 3 z(x Ax B)
8 eleq1 2097 . . . 4 (z = x → (z Ax A))
9 eleq1 2097 . . . 4 (z = x → (z Bx B))
108, 9imbi12d 223 . . 3 (z = x → ((z Az B) ↔ (x Ax B)))
116, 7, 10cbval 1634 . 2 (z(z Az B) ↔ x(x Ax B))
121, 11bitri 173 1 (ABx(x Ax B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   ∈ wcel 1390  Ⅎwnfc 2162   ⊆ 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:  dfss3f  2931  ssrd  2944  ss2ab  3002
 Copyright terms: Public domain W3C validator