Theorem eqif 4126

Description: Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)

Assertion

Ref	Expression
eqif	|- ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) )

Proof of Theorem eqif

Step	Hyp	Ref	Expression
1		eqeq2 2633	. 2 |- ( if ( ph , B , C ) = B -> ( A = if ( ph , B , C ) <-> A = B ) )
2		eqeq2 2633	. 2 |- ( if ( ph , B , C ) = C -> ( A = if ( ph , B , C ) <-> A = C ) )
3	1, 2	elimif 4122	1 |- ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  ifval  4127  xpima  5576  fin23lem19  9158  fin23lem28  9162  fin23lem29  9163  fin23lem30  9164  aalioulem3  24089  iocinif  29543  fsumcvg4  29996  ind1a  30081  esumsnf  30126  itg2addnclem2  33462  clsk1indlem4  38342  afvpcfv0  41226