Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rabeq12OLD Unicode version

Theorem rabeq12OLD 26350
Description: Equality of restricted class abstractions. (Moved to rabeqbidv 2783 in main set.mm and may be deleted by mathbox owner, JM. --NM 15-Sep-2011.) (Contributed by Jeff Madsen, 1-Dec-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
rabeq12.1OLD  |-  ( ph  ->  A  =  B )
rabeq12.2OLD  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
rabeq12OLD  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem rabeq12OLD
StepHypRef Expression
1 rabeq12.1OLD . 2  |-  ( ph  ->  A  =  B )
2 rabeq12.2OLD . 2  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2rabeqbidv 2783 1  |-  ( ph  ->  { x  e.  A  |  ps }  =  {
x  e.  B  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   {crab 2547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552
  Copyright terms: Public domain W3C validator