Theorem pw1if 7442

Description: Expressing a truth value in terms of an if expression. (Contributed by Jim Kingdon, 10-Jan-2026.)

Assertion

Ref	Expression
pw1if	|- ( A e. ~P 1o -> if ( A = 1o , 1o , (/) ) = A )

Proof of Theorem pw1if

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 |- ( ( A e. ~P 1o /\ x e. if ( A = 1o , 1o , (/) ) ) -> x e. if ( A = 1o , 1o , (/) ) )
2		elif 3617	. . . . . . 7 |- ( x e. if ( A = 1o , 1o , (/) ) <-> ( ( A = 1o /\ x e. 1o ) \/ ( -. A = 1o /\ x e. (/) ) ) )
3		noel 3498	. . . . . . . . 9 |- -. x e. (/)
4	3	intnan 936	. . . . . . . 8 |- -. ( -. A = 1o /\ x e. (/) )
5	4	biorfi 753	. . . . . . 7 |- ( ( A = 1o /\ x e. 1o ) <-> ( ( A = 1o /\ x e. 1o ) \/ ( -. A = 1o /\ x e. (/) ) ) )
6	2, 5	bitr4i 187	. . . . . 6 |- ( x e. if ( A = 1o , 1o , (/) ) <-> ( A = 1o /\ x e. 1o ) )
7	1, 6	sylib 122	. . . . 5 |- ( ( A e. ~P 1o /\ x e. if ( A = 1o , 1o , (/) ) ) -> ( A = 1o /\ x e. 1o ) )
8	7	simprd 114	. . . 4 |- ( ( A e. ~P 1o /\ x e. if ( A = 1o , 1o , (/) ) ) -> x e. 1o )
9	7	simpld 112	. . . 4 |- ( ( A e. ~P 1o /\ x e. if ( A = 1o , 1o , (/) ) ) -> A = 1o )
10	8, 9	eleqtrrd 2311	. . 3 |- ( ( A e. ~P 1o /\ x e. if ( A = 1o , 1o , (/) ) ) -> x e. A )
11		elex2 2819	. . . . 5 |- ( x e. A -> E. y y e. A )
12		pw1m 7441	. . . . 5 |- ( ( A e. ~P 1o /\ E. y y e. A ) -> A = 1o )
13	11, 12	sylan2 286	. . . 4 |- ( ( A e. ~P 1o /\ x e. A ) -> A = 1o )
14		simpr 110	. . . . 5 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. A )
15	14, 13	eleqtrd 2310	. . . 4 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. 1o )
16	13, 15, 6	sylanbrc 417	. . 3 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. if ( A = 1o , 1o , (/) ) )
17	10, 16	impbida 600	. 2 |- ( A e. ~P 1o -> ( x e. if ( A = 1o , 1o , (/) ) <-> x e. A ) )
18	17	eqrdv 2229	1 |- ( A e. ~P 1o -> if ( A = 1o , 1o , (/) ) = A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   E.wex 1540   e. wcel 2202   (/)c0 3494   ifcif 3605   ~Pcpw 3652   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

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-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-suc 4468  df-1o 6581

This theorem is referenced by:  pw1map  16596