Theorem iftrueb01 7440

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 6607	. . . 4 |- (/) e. 1o
2		elex2 2819	. . . 4 |- ( (/) e. 1o -> E. x x e. 1o )
3	1, 2	ax-mp 5	. . 3 |- E. x x e. 1o
4		eleq2 2295	. . . . . 6 |- ( if ( ph , 1o , (/) ) = 1o -> ( x e. if ( ph , 1o , (/) ) <-> x e. 1o ) )
5		elif 3617	. . . . . . 7 |- ( x e. if ( ph , 1o , (/) ) <-> ( ( ph /\ x e. 1o ) \/ ( -. ph /\ x e. (/) ) ) )
6		noel 3498	. . . . . . . . 9 |- -. x e. (/)
7	6	intnan 936	. . . . . . . 8 |- -. ( -. ph /\ x e. (/) )
8	7	biorfi 753	. . . . . . 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 1867	. . 3 |- ( if ( ph , 1o , (/) ) = 1o -> ( E. x x e. 1o -> ph ) )
14	3, 13	mpi 15	. 2 |- ( if ( ph , 1o , (/) ) = 1o -> ph )
15		iftrue 3610	. 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 715   = wceq 1397   E.wex 1540   e. wcel 2202   (/)c0 3494   ifcif 3605   1oc1o 6574

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-if 3606  df-sn 3675  df-suc 4468  df-1o 6581

This theorem is referenced by:  pw1map  16596