Theorem eqifdc 3539

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 822	. . 3 |- (DECID ph -> ( ph \/ -. ph ) )
2		simpr 109	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ ph ) -> ph )
3		simpl 108	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ ph ) -> A = if ( ph , B , C ) )
4	2	iftrued 3512	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ ph ) -> if ( ph , B , C ) = B )
5	3, 4	eqtrd 2190	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ ph ) -> A = B )
6	2, 5	jca 304	. . . . 5 |- ( ( A = if ( ph , B , C ) /\ ph ) -> ( ph /\ A = B ) )
7	6	ex 114	. . . 4 |- ( A = if ( ph , B , C ) -> ( ph -> ( ph /\ A = B ) ) )
8		simpr 109	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> -. ph )
9		simpl 108	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> A = if ( ph , B , C ) )
10	8	iffalsed 3515	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> if ( ph , B , C ) = C )
11	9, 10	eqtrd 2190	. . . . . 6 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> A = C )
12	8, 11	jca 304	. . . . 5 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> ( -. ph /\ A = C ) )
13	12	ex 114	. . . 4 |- ( A = if ( ph , B , C ) -> ( -. ph -> ( -. ph /\ A = C ) ) )
14	7, 13	orim12d 776	. . 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 109	. . . 4 |- ( ( ph /\ A = B ) -> A = B )
17		simpl 108	. . . . 5 |- ( ( ph /\ A = B ) -> ph )
18	17	iftrued 3512	. . . 4 |- ( ( ph /\ A = B ) -> if ( ph , B , C ) = B )
19	16, 18	eqtr4d 2193	. . 3 |- ( ( ph /\ A = B ) -> A = if ( ph , B , C ) )
20		simpr 109	. . . 4 |- ( ( -. ph /\ A = C ) -> A = C )
21		simpl 108	. . . . 5 |- ( ( -. ph /\ A = C ) -> -. ph )
22	21	iffalsed 3515	. . . 4 |- ( ( -. ph /\ A = C ) -> if ( ph , B , C ) = C )
23	20, 22	eqtr4d 2193	. . 3 |- ( ( -. ph /\ A = C ) -> A = if ( ph , B , C ) )
24	19, 23	jaoi 706	. 2 |- ( ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) -> A = if ( ph , B , C ) )
25	15, 24	impbid1 141	1 |- (DECID ph -> ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   = wceq 1335   ifcif 3505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-dc 821  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-if 3506

This theorem is referenced by:  fodjum  7083  xrmaxiflemcom  11139  subctctexmid  13544