Theorem elimif 4122

Description: Elimination of a conditional operator contained in a wff ps. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)

Hypotheses

Ref	Expression
elimif.1	|- ( if ( ph , A , B ) = A -> ( ps <-> ch ) )
elimif.2	|- ( if ( ph , A , B ) = B -> ( ps <-> th ) )

Assertion

Ref	Expression
elimif	|- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )

Proof of Theorem elimif

Step	Hyp	Ref	Expression
1		iftrue 4092	. . 3 |- ( ph -> if ( ph , A , B ) = A )
2		elimif.1	. . 3 |- ( if ( ph , A , B ) = A -> ( ps <-> ch ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( ps <-> ch ) )
4		iffalse 4095	. . 3 |- ( -. ph -> if ( ph , A , B ) = B )
5		elimif.2	. . 3 |- ( if ( ph , A , B ) = B -> ( ps <-> th ) )
6	4, 5	syl 17	. 2 |- ( -. ph -> ( ps <-> th ) )
7	3, 6	cases 992	1 |- ( ps <-> ( ( ph /\ ch ) \/ ( -. ph /\ th ) ) )

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:  eqif  4126  elif  4128  ifel  4129  ftc1anclem5  33489  clsk1indlem2  38340