Theorem ifval 4127

Description: Another expression of the value of the if predicate, analogous to eqif 4126. See also the more specialized iftrue 4092 and iffalse 4095. (Contributed by BJ, 6-Apr-2019.)

Assertion

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

Proof of Theorem ifval

Step	Hyp	Ref	Expression
1		eqif 4126	. 2 |- ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) )
2		cases2 993	. 2 |- ( ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) <-> ( ( ph -> A = B ) /\ ( -. ph -> A = C ) ) )
3	1, 2	bitri 264	1 |- ( A = if ( ph , B , C ) <-> ( ( ph -> A = B ) /\ ( -. ph -> A = C ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> 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:  dfiota4  5879  bj-projval  32984