Theorem exmidcon 16950

Description: Excluded middle is equivalent to the form of contraposition which removes negation. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromcon 1498. (Contributed by Jim Kingdon, 22-Apr-2026.)

Assertion

Ref	Expression
exmidcon	|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )

Distinct variable group:   x, y

Proof of Theorem exmidcon

Step	Hyp	Ref	Expression
1		exmidexmid 4328	. . . . 5 |- (EXMID -> DECID y = 1o )
2		condc 865	. . . . 5 |- (DECID y = 1o -> ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )
3	1, 2	syl 14	. . . 4 |- (EXMID -> ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )
4	3	ralrimivw 2624	. . 3 |- (EXMID -> A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )
5	4	ralrimivw 2624	. 2 |- (EXMID -> A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )
6		eqeq1 2245	. . . . . . . 8 |- ( x = 1o -> ( x = 1o <-> 1o = 1o ) )
7	6	notbid 677	. . . . . . 7 |- ( x = 1o -> ( -. x = 1o <-> -. 1o = 1o ) )
8	7	imbi2d 230	. . . . . 6 |- ( x = 1o -> ( ( -. y = 1o -> -. x = 1o ) <-> ( -. y = 1o -> -. 1o = 1o ) ) )
9	6	imbi1d 231	. . . . . 6 |- ( x = 1o -> ( ( x = 1o -> y = 1o ) <-> ( 1o = 1o -> y = 1o ) ) )
10	8, 9	imbi12d 234	. . . . 5 |- ( x = 1o -> ( ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) <-> ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) ) )
11	10	ralbidv 2550	. . . 4 |- ( x = 1o -> ( A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) <-> A. y e. ~P 1o ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) ) )
12		id 19	. . . 4 |- ( A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) -> A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )
13		1oex 6685	. . . . . 6 |- 1o e. _V
14	13	pwid 3703	. . . . 5 |- 1o e. ~P 1o
15	14	a1i 9	. . . 4 |- ( A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) -> 1o e. ~P 1o )
16	11, 12, 15	rspcdva 2934	. . 3 |- ( A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) -> A. y e. ~P 1o ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) )
17		eqid 2238	. . . . . . 7 |- 1o = 1o
18		ax-in2 624	. . . . . . . . 9 |- ( -. -. y = 1o -> ( -. y = 1o -> -. 1o = 1o ) )
19	18	adantr 276	. . . . . . . 8 |- ( ( -. -. y = 1o /\ ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) ) -> ( -. y = 1o -> -. 1o = 1o ) )
20		simpr 110	. . . . . . . 8 |- ( ( -. -. y = 1o /\ ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) ) -> ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) )
21	19, 20	mpd 13	. . . . . . 7 |- ( ( -. -. y = 1o /\ ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) ) -> ( 1o = 1o -> y = 1o ) )
22	17, 21	mpi 15	. . . . . 6 |- ( ( -. -. y = 1o /\ ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) ) -> y = 1o )
23	22	expcom 116	. . . . 5 |- ( ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) -> ( -. -. y = 1o -> y = 1o ) )
24	23	ralimi 2613	. . . 4 |- ( A. y e. ~P 1o ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) -> A. y e. ~P 1o ( -. -. y = 1o -> y = 1o ) )
25		exmidnotnotr 16949	. . . 4 |- (EXMID <-> A. y e. ~P 1o ( -. -. y = 1o -> y = 1o ) )
26	24, 25	sylibr 134	. . 3 |- ( A. y e. ~P 1o ( ( -. y = 1o -> -. 1o = 1o ) -> ( 1o = 1o -> y = 1o ) ) -> EXMID )
27	16, 26	syl 14	. 2 |- ( A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) -> EXMID )
28	5, 27	impbii 126	1 |- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   ~Pcpw 3685  EXMIDwem 4326   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-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-exmid 4327  df-iord 4506  df-on 4508  df-suc 4511  df-1o 6677

This theorem is referenced by: (None)