Theorem ifeq12 3365

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

Step	Hyp	Ref	Expression
1		ifeq1 3354	. 2 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
2		ifeq2 3355	. 2 |- ( C = D -> if ( ph , B , C ) = if ( ph , B , D ) )
3	1, 2	sylan9eq 2133	1 |- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   ifcif 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-un 2977  df-if 3352

This theorem is referenced by: (None)