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

Theorem ifeq1d 3539
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 3529 . 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 3525
This theorem is referenced by:  ifeq12d  3541  ifeq1da  3550  cantnflem1d  7344  cantnflem1  7345  isumless  12252  subgmulg  14583  gsumzsplit  15154  evlslem2  16197  cnmpt2pc  18374  pcoval2  18462  pcopt  18468  itgz  19083  iblss2  19108  itgss  19114  itgcn  19145  plyeq0lem  19540  dgrcolem2  19603  plydivlem4  19624  leibpi  20186  chtublem  20398  sumdchr  20459  bposlem6  20476  lgsval  20487  dchrvmasumiflem2  20599  padicabvcxp  20729  dfrdg3  23508  linevala2  25431  sgplpte21  25485  sgplpte22  25491  isray2  25506  isray  25507
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rab 2525  df-v 2759  df-un 3118  df-if 3526
  Copyright terms: Public domain W3C validator