Theorem iftrueb01 7404

Description: Using an if expression to represent a truth value by (/) or 1o. Unlike some theorems using if, ph does not need to be decidable. (Contributed by Jim Kingdon, 9-Jan-2026.)

Assertion

Ref	Expression
iftrueb01	|- ( if ( ph , 1o , (/) ) = 1o <-> ph )

Proof of Theorem iftrueb01

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1o 6584	. . . 4 |- (/) e. 1o
2		elex2 2816	. . . 4 |- ( (/) e. 1o -> E. x x e. 1o )
3	1, 2	ax-mp 5	. . 3 |- E. x x e. 1o
4		eleq2 2293	. . . . . 6 |- ( if ( ph , 1o , (/) ) = 1o -> ( x e. if ( ph , 1o , (/) ) <-> x e. 1o ) )
5		elif 3614	. . . . . . 7 |- ( x e. if ( ph , 1o , (/) ) <-> ( ( ph /\ x e. 1o ) \/ ( -. ph /\ x e. (/) ) ) )
6		noel 3495	. . . . . . . . 9 |- -. x e. (/)
7	6	intnan 934	. . . . . . . 8 |- -. ( -. ph /\ x e. (/) )
8	7	biorfi 751	. . . . . . 7 |- ( ( ph /\ x e. 1o ) <-> ( ( ph /\ x e. 1o ) \/ ( -. ph /\ x e. (/) ) ) )
9	5, 8	bitr4i 187	. . . . . 6 |- ( x e. if ( ph , 1o , (/) ) <-> ( ph /\ x e. 1o ) )
10	4, 9	bitr3di 195	. . . . 5 |- ( if ( ph , 1o , (/) ) = 1o -> ( x e. 1o <-> ( ph /\ x e. 1o ) ) )
11		pm4.71r 390	. . . . 5 |- ( ( x e. 1o -> ph ) <-> ( x e. 1o <-> ( ph /\ x e. 1o ) ) )
12	10, 11	sylibr 134	. . . 4 |- ( if ( ph , 1o , (/) ) = 1o -> ( x e. 1o -> ph ) )
13	12	exlimdv 1865	. . 3 |- ( if ( ph , 1o , (/) ) = 1o -> ( E. x x e. 1o -> ph ) )
14	3, 13	mpi 15	. 2 |- ( if ( ph , 1o , (/) ) = 1o -> ph )
15		iftrue 3607	. 2 |- ( ph -> if ( ph , 1o , (/) ) = 1o )
16	14, 15	impbii 126	1 |- ( if ( ph , 1o , (/) ) = 1o <-> ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   (/)c0 3491   ifcif 3602   1oc1o 6553

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4209

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-if 3603  df-sn 3672  df-suc 4461  df-1o 6560

This theorem is referenced by:  pw1map  16320