Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  freq12d Unicode version

Theorem freq12d 27238
Description: Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
weeq12d.l  |-  ( ph  ->  R  =  S )
weeq12d.r  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
freq12d  |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  B ) )

Proof of Theorem freq12d
StepHypRef Expression
1 weeq12d.l . . 3  |-  ( ph  ->  R  =  S )
2 freq1 4379 . . 3  |-  ( R  =  S  ->  ( R  Fr  A  <->  S  Fr  A ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  A ) )
4 weeq12d.r . . 3  |-  ( ph  ->  A  =  B )
5 freq2 4380 . . 3  |-  ( A  =  B  ->  ( S  Fr  A  <->  S  Fr  B ) )
64, 5syl 15 . 2  |-  ( ph  ->  ( S  Fr  A  <->  S  Fr  B ) )
73, 6bitrd 244 1  |-  ( ph  ->  ( R  Fr  A  <->  S  Fr  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    Fr wfr 4365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-rex 2562  df-in 3172  df-ss 3179  df-br 4040  df-fr 4368
  Copyright terms: Public domain W3C validator