ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeq2d Unicode version

Theorem opeq2d 3597
Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
opeq2d  |-  ( ph  -> 
<. C ,  A >.  = 
<. C ,  B >. )

Proof of Theorem opeq2d
StepHypRef Expression
1 opeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 opeq2 3591 . 2  |-  ( A  =  B  ->  <. C ,  A >.  =  <. C ,  B >. )
31, 2syl 14 1  |-  ( ph  -> 
<. C ,  A >.  = 
<. C ,  B >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   <.cop 3419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425
This theorem is referenced by:  tfr1onlemaccex  6018  tfrcllemaccex  6031  fundmen  6375  recexnq  6712  elreal2  7131  frecuzrdgrrn  9560  frec2uzrdg  9561  frecuzrdgrcl  9562  frecuzrdgsuc  9566  frecuzrdgrclt  9567  frecuzrdgg  9568  frecuzrdgsuctlem  9575  iseqeq2  9595  iseqeq3  9596  iseqval  9600  iseqvalcbv  9601  iseqvalt  9602  eucalgval  10661
  Copyright terms: Public domain W3C validator