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Theorem ifeq1da 3707
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
ifeq1da.1  |-  ( (
ph  /\  ps )  ->  A  =  B )
Assertion
Ref Expression
ifeq1da  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)

Proof of Theorem ifeq1da
StepHypRef Expression
1 ifeq1da.1 . . 3  |-  ( (
ph  /\  ps )  ->  A  =  B )
21ifeq1d 3696 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
3 iffalse 3689 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  A ,  C
)  =  C )
4 iffalse 3689 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  B ,  C
)  =  C )
53, 4eqtr4d 2422 . . 3  |-  ( -. 
ps  ->  if ( ps ,  A ,  C
)  =  if ( ps ,  B ,  C ) )
65adantl 453 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
72, 6pm2.61dan 767 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649   ifcif 3682
This theorem is referenced by:  cantnflem1d  7577  cantnflem1  7578  dfac12lem1  7956  xrmaxeq  10699  xrmineq  10700  rexmul  10782  max0add  12042  fsumser  12451  ramcl  13324  dmdprdsplitlem  15522  coe1pwmul  16598  ptcld  17566  copco  18914  ibllem  19523  itgvallem3  19544  iblposlem  19550  iblss2  19564  iblmulc2  19589  cnplimc  19641  limcco  19647  dvexp3  19729  dchrinvcl  20904  lgsval2lem  20957  lgsval4lem  20958  lgsneg  20970  lgsmod  20972  lgsdilem2  20982  rpvmasum2  21073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-un 3268  df-if 3683
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