Theorem ifeq12 3542

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 3529	. 2 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
2		ifeq2 3530	. 2 |- ( C = D -> if ( ph , B , C ) = if ( ph , B , D ) )
3	1, 2	sylan9eq 2223	1 |- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   ifcif 3526

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-if 3527

This theorem is referenced by:  xaddmnf1  9805