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

Theorem dfss2f 3129
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  |-  F/_ x A
dfss2f.2  |-  F/_ x B
Assertion
Ref Expression
dfss2f  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )

Proof of Theorem dfss2f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfss2 3127 . 2  |-  ( A 
C_  B  <->  A. z
( z  e.  A  ->  z  e.  B ) )
2 dfss2f.1 . . . . 5  |-  F/_ x A
32nfcri 2300 . . . 4  |-  F/ x  z  e.  A
4 dfss2f.2 . . . . 5  |-  F/_ x B
54nfcri 2300 . . . 4  |-  F/ x  z  e.  B
63, 5nfim 1559 . . 3  |-  F/ x
( z  e.  A  ->  z  e.  B )
7 nfv 1515 . . 3  |-  F/ z ( x  e.  A  ->  x  e.  B )
8 eleq1 2227 . . . 4  |-  ( z  =  x  ->  (
z  e.  A  <->  x  e.  A ) )
9 eleq1 2227 . . . 4  |-  ( z  =  x  ->  (
z  e.  B  <->  x  e.  B ) )
108, 9imbi12d 233 . . 3  |-  ( z  =  x  ->  (
( z  e.  A  ->  z  e.  B )  <-> 
( x  e.  A  ->  x  e.  B ) ) )
116, 7, 10cbval 1741 . 2  |-  ( A. z ( z  e.  A  ->  z  e.  B )  <->  A. x
( x  e.  A  ->  x  e.  B ) )
121, 11bitri 183 1  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1340    e. wcel 2135   F/_wnfc 2293    C_ wss 3112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-in 3118  df-ss 3125
This theorem is referenced by:  dfss3f  3130  ssrd  3143  ssrmof  3201  ss2ab  3206
  Copyright terms: Public domain W3C validator