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Theorem ifeq12 3550
Description: Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
Assertion
Ref Expression
ifeq12  |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D )
)

Proof of Theorem ifeq12
StepHypRef Expression
1 ifeq1 3537 . 2  |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C ) )
2 ifeq2 3538 . 2  |-  ( C  =  D  ->  if ( ph ,  B ,  C )  =  if ( ph ,  B ,  D ) )
31, 2sylan9eq 2230 1  |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   ifcif 3534
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-un 3133  df-if 3535
This theorem is referenced by:  xaddmnf1  9835
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