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Theorem ifeq12 3344
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 3334 . 2  |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C ) )
2 ifeq2 3335 . 2  |-  ( C  =  D  ->  if ( ph ,  B ,  C )  =  if ( ph ,  B ,  D ) )
31, 2sylan9eq 2092 1  |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   ifcif 3331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-if 3332
This theorem is referenced by: (None)
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