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Theorem ifeq12d 3568
Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
Hypotheses
Ref Expression
ifeq1d.1  |-  ( ph  ->  A  =  B )
ifeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
ifeq12d  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D )
)

Proof of Theorem ifeq12d
StepHypRef Expression
1 ifeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21ifeq1d 3566 . 2  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
3 ifeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43ifeq2d 3567 . 2  |-  ( ph  ->  if ( ps ,  B ,  C )  =  if ( ps ,  B ,  D )
)
52, 4eqtrd 2222 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364   ifcif 3549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-un 3148  df-if 3550
This theorem is referenced by:  ifbieq12d  3575  xaddpnf1  9864  exp3val  10540  eucalgval  12072  ennnfonelemp1  12425  ennnfonelemnn0  12441  mulgfvalg  13029  mulgpropdg  13070  lgsval  14802
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