Theorem eqifdc 3501

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 821	. . 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 3476	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ ph ) -> if ( ph , B , C ) = B )
5	3, 4	eqtrd 2170	. . . . . 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 3479	. . . . . . 7 |- ( ( A = if ( ph , B , C ) /\ -. ph ) -> if ( ph , B , C ) = C )
11	9, 10	eqtrd 2170	. . . . . 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 775	. . 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 3476	. . . 4 |- ( ( ph /\ A = B ) -> if ( ph , B , C ) = B )
19	16, 18	eqtr4d 2173	. . 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 3479	. . . 4 |- ( ( -. ph /\ A = C ) -> if ( ph , B , C ) = C )
23	20, 22	eqtr4d 2173	. . 3 |- ( ( -. ph /\ A = C ) -> A = if ( ph , B , C ) )
24	19, 23	jaoi 705	. 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 697  DECID wdc 819   = wceq 1331   ifcif 3469

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 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-if 3470

This theorem is referenced by:  fodjum  7011  xrmaxiflemcom  11011  subctctexmid  13185