Theorem eqifdc 3616

Description: Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)

Assertion

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

Proof of Theorem eqifdc

Step	Hyp	Ref	Expression
1		exmiddc 838	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
2		simpr 110	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ ph ) -> ph )
3		simpl 109	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ ph ) -> A = if ( ph , B , C ) )
4	2	iftrued 3586	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ ph ) -> if ( ph , B , C ) = B )
5	3, 4	eqtrd 2240	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ ph ) -> A = B )
6	2, 5	jca 306	. . . . 5 |- ( ( A = if ( ph , B , C ) /\ ph ) -> ( ph /\ A = B ) )
7	6	ex 115	. . . 4 |- ( A = if ( ph , B , C ) -> ( ph -> ( ph /\ A = B ) ) )
8		simpr 110	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> -. ph )
9		simpl 109	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> A = if ( ph , B , C ) )
10	8	iffalsed 3589	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> if ( ph , B , C ) = C )
11	9, 10	eqtrd 2240	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> A = C )
12	8, 11	jca 306	. . . . 5 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> ( -. ph /\ A = C ) )
13	12	ex 115	. . . 4 |- ( A = if ( ph , B , C ) -> ( -. ph -> ( -. ph /\ A = C ) ) )
14	7, 13	orim12d 788	. . 3 |- ( A = if ( ph , B , C ) -> ( ( ph \/ -. ph ) -> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )
15	1, 14	syl5com 29	. 2 |- (DECID ph -> ( A = if ( ph , B , C ) -> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )
16		simpr 110	. . . 4 |- ( ( ph /\ A = B ) -> A = B )
17		simpl 109	. . . . 5 |- ( ( ph /\ A = B ) -> ph )
18	17	iftrued 3586	. . . 4 |- ( ( ph /\ A = B ) -> if ( ph , B , C ) = B )
19	16, 18	eqtr4d 2243	. . 3 |- ( ( ph /\ A = B ) -> A = if ( ph , B , C ) )
20		simpr 110	. . . 4 |- ( ( -. ph /\ A = C ) -> A = C )
21		simpl 109	. . . . 5 |- ( ( -. ph /\ A = C ) -> -. ph )
22	21	iffalsed 3589	. . . 4 |- ( ( -. ph /\ A = C ) -> if ( ph , B , C ) = C )
23	20, 22	eqtr4d 2243	. . 3 |- ( ( -. ph /\ A = C ) -> A = if ( ph , B , C ) )
24	19, 23	jaoi 718	. 2 |- ( ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) -> A = if ( ph , B , C ) )
25	15, 24	impbid1 142	1 |- (DECID ph -> ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710  DECID wdc 836   = wceq 1373   ifcif 3579

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-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-dc 837  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-if 3580

This theorem is referenced by:  ifnebibdc  3625  fodjum  7274  nninfwlporlemd  7300  xrmaxiflemcom  11675  gsumfzval  13338  subctctexmid  16139