Theorem pw1if 7574

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 3649	. . . . . . 7 |- ( x e. if ( A = 1o , 1o , (/) ) <-> ( ( A = 1o /\ x e. 1o ) \/ ( -. A = 1o /\ x e. (/) ) ) )
3		noel 3525	. . . . . . . . 9 |- -. x e. (/)
4	3	intnan 941	. . . . . . . 8 |- -. ( -. A = 1o /\ x e. (/) )
5	4	biorfi 758	. . . . . . 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 2318	. . 3 |- ( ( A e. ~P 1o /\ x e. if ( A = 1o , 1o , (/) ) ) -> x e. A )
11		elex2 2838	. . . . 5 |- ( x e. A -> E. y y e. A )
12		pw1m 7573	. . . . 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 2317	. . . 4 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. 1o )
16	13, 15, 6	sylanbrc 421	. . 3 |- ( ( A e. ~P 1o /\ x e. A ) -> x e. if ( A = 1o , 1o , (/) ) )
17	10, 16	impbida 604	. 2 |- ( A e. ~P 1o -> ( x e. if ( A = 1o , 1o , (/) ) <-> x e. A ) )
18	17	eqrdv 2236	1 |- ( A e. ~P 1o -> if ( A = 1o , 1o , (/) ) = A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   (/)c0 3520   ifcif 3635   ~Pcpw 3685   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

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-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-suc 4511  df-1o 6677

This theorem is referenced by:  pw1map  16939