Theorem ifeq12 3536

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 3523	. 2 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
2		ifeq2 3524	. 2 |- ( C = D -> if ( ph , B , C ) = if ( ph , B , D ) )
3	1, 2	sylan9eq 2219	1 |- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rab 2453  df-v 2728  df-un 3120  df-if 3521

This theorem is referenced by:  xaddmnf1  9784