Theorem ifeq12 3622

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 3608	. 2 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
2		ifeq2 3609	. 2 |- ( C = D -> if ( ph , B , C ) = if ( ph , B , D ) )
3	1, 2	sylan9eq 2284	1 |- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   ifcif 3605

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-un 3204  df-if 3606

This theorem is referenced by:  xaddmnf1  10082