ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifeq2d Unicode version

Theorem ifeq2d 3485
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
Hypothesis
Ref Expression
ifeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ifeq2d  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)

Proof of Theorem ifeq2d
StepHypRef Expression
1 ifeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ifeq2 3473 . 2  |-  ( A  =  B  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
31, 2syl 14 1  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   ifcif 3469
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-un 3070  df-if 3470
This theorem is referenced by:  ifeq12d  3486  ifbieq2d  3491  exp3val  10288  xrmaxiflemcom  11011  peano4nninf  13185  peano3nninf  13186
  Copyright terms: Public domain W3C validator