MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifeq1d Unicode version

Theorem ifeq1d 3484
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
Hypothesis
Ref Expression
ifeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ifeq1d  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)

Proof of Theorem ifeq1d
StepHypRef Expression
1 ifeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ifeq1 3474 . 2  |-  ( A  =  B  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
31, 2syl 17 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619   ifcif 3470
This theorem is referenced by:  ifeq12d  3486  ifeq1da  3495  cantnflem1d  7274  cantnflem1  7275  isumless  12178  subgmulg  14470  gsumzsplit  15041  evlslem2  16081  cnmpt2pc  18258  pcoval2  18346  pcopt  18352  itgz  18967  iblss2  18992  itgss  18998  itgcn  19029  plyeq0lem  19424  dgrcolem2  19487  plydivlem4  19508  leibpi  20070  chtublem  20282  sumdchr  20343  bposlem6  20360  lgsval  20371  dchrvmasumiflem2  20483  padicabvcxp  20613  dfrdg3  23321  linevala2  25244  sgplpte21  25298  sgplpte22  25304  isray2  25319  isray  25320
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-v 2729  df-un 3083  df-if 3471
  Copyright terms: Public domain W3C validator