Theorem ifeq12 3573

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 3560	. 2 |- ( A = B -> if ( ph , A , C ) = if ( ph , B , C ) )
2		ifeq2 3561	. 2 |- ( C = D -> if ( ph , B , C ) = if ( ph , B , D ) )
3	1, 2	sylan9eq 2246	1 |- ( ( A = B /\ C = D ) -> if ( ph , A , C ) = if ( ph , B , D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   ifcif 3557

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-un 3157  df-if 3558

This theorem is referenced by:  xaddmnf1  9914