Theorem iftrueb01 7572

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 6703	. . . 4 |- (/) e. 1o
2		elex2 2838	. . . 4 |- ( (/) e. 1o -> E. x x e. 1o )
3	1, 2	ax-mp 5	. . 3 |- E. x x e. 1o
4		eleq2 2302	. . . . . 6 |- ( if ( ph , 1o , (/) ) = 1o -> ( x e. if ( ph , 1o , (/) ) <-> x e. 1o ) )
5		elif 3649	. . . . . . 7 |- ( x e. if ( ph , 1o , (/) ) <-> ( ( ph /\ x e. 1o ) \/ ( -. ph /\ x e. (/) ) ) )
6		noel 3525	. . . . . . . . 9 |- -. x e. (/)
7	6	intnan 941	. . . . . . . 8 |- -. ( -. ph /\ x e. (/) )
8	7	biorfi 758	. . . . . . 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 394	. . . . 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 1872	. . 3 |- ( if ( ph , 1o , (/) ) = 1o -> ( E. x x e. 1o -> ph ) )
14	3, 13	mpi 15	. 2 |- ( if ( ph , 1o , (/) ) = 1o -> ph )
15		iftrue 3642	. 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 720   = wceq 1402   E.wex 1545   e. wcel 2209   (/)c0 3520   ifcif 3635   1oc1o 6670

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-if 3636  df-sn 3711  df-suc 4511  df-1o 6677

This theorem is referenced by:  pw1map  16939