ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqeqan12rd Unicode version

Theorem eqeqan12rd 2056
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
Hypotheses
Ref Expression
eqeqan12rd.1  |-  ( ph  ->  A  =  B )
eqeqan12rd.2  |-  ( ps 
->  C  =  D
)
Assertion
Ref Expression
eqeqan12rd  |-  ( ( ps  /\  ph )  ->  ( A  =  C  <-> 
B  =  D ) )

Proof of Theorem eqeqan12rd
StepHypRef Expression
1 eqeqan12rd.1 . . 3  |-  ( ph  ->  A  =  B )
2 eqeqan12rd.2 . . 3  |-  ( ps 
->  C  =  D
)
31, 2eqeqan12d 2055 . 2  |-  ( (
ph  /\  ps )  ->  ( A  =  C  <-> 
B  =  D ) )
43ancoms 255 1  |-  ( ( ps  /\  ph )  ->  ( A  =  C  <-> 
B  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243
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-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator