Metamath Proof Explorer


Theorem ich2exprop

Description: If the setvar variables are interchangeable in a wff, there is an ordered pair fulfilling the wff iff there is an unordered pair fulfilling the wff. (Contributed by AV, 16-Jul-2023)

Ref Expression
Assertion ich2exprop A X B X a b φ a b A B = a b φ a b A B = a b φ

Proof

Step Hyp Ref Expression
1 nfv a A X
2 nfv a B X
3 nfich1 a a b φ
4 1 2 3 nf3an a A X B X a b φ
5 nfv a A B = x y
6 nfcv _ a y
7 nfsbc1v a [˙x / a]˙ φ
8 6 7 nfsbcw a [˙y / b]˙ [˙x / a]˙ φ
9 5 8 nfan a A B = x y [˙y / b]˙ [˙x / a]˙ φ
10 9 nfex a y A B = x y [˙y / b]˙ [˙x / a]˙ φ
11 10 nfex a x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
12 nfv b A X
13 nfv b B X
14 nfich2 b a b φ
15 12 13 14 nf3an b A X B X a b φ
16 nfv b A B = x y
17 nfsbc1v b [˙y / b]˙ [˙x / a]˙ φ
18 16 17 nfan b A B = x y [˙y / b]˙ [˙x / a]˙ φ
19 18 nfex b y A B = x y [˙y / b]˙ [˙x / a]˙ φ
20 19 nfex b x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
21 vex a V
22 vex b V
23 preq12bg A X B X a V b V A B = a b A = a B = b A = b B = a
24 21 22 23 mpanr12 A X B X A B = a b A = a B = b A = b B = a
25 24 3adant3 A X B X a b φ A B = a b A = a B = b A = b B = a
26 or2expropbilem1 A X B X A = a B = b φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
27 26 3adant3 A X B X a b φ A = a B = b φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
28 ichcom a b φ b a φ
29 28 biimpi a b φ b a φ
30 29 3ad2ant3 A X B X a b φ b a φ
31 30 adantr A X B X a b φ φ b a φ
32 22 21 pm3.2i b V a V
33 32 a1i A = b B = a b V a V
34 31 33 anim12i A X B X a b φ φ A = b B = a b a φ b V a V
35 simpr A X B X a b φ φ φ
36 opeq12 A = b B = a A B = b a
37 35 36 anim12ci A X B X a b φ φ A = b B = a A B = b a φ
38 nfv x A B = b a φ
39 nfv y A B = b a φ
40 opeq12 x = b y = a x y = b a
41 40 eqeq2d x = b y = a A B = x y A B = b a
42 41 adantl b a φ x = b y = a A B = x y A B = b a
43 dfsbcq y = a [˙y / b]˙ [˙x / a]˙ φ [˙a / b]˙ [˙x / a]˙ φ
44 43 adantl x = b y = a [˙y / b]˙ [˙x / a]˙ φ [˙a / b]˙ [˙x / a]˙ φ
45 44 adantl b a φ x = b y = a [˙y / b]˙ [˙x / a]˙ φ [˙a / b]˙ [˙x / a]˙ φ
46 sbceq1a x = b [˙a / b]˙ [˙x / a]˙ φ [˙b / x]˙ [˙a / b]˙ [˙x / a]˙ φ
47 46 adantr x = b y = a [˙a / b]˙ [˙x / a]˙ φ [˙b / x]˙ [˙a / b]˙ [˙x / a]˙ φ
48 df-ich b a φ b a b x a b x a φ φ
49 sbsbc b x a b x a φ [˙b / x]˙ a b x a φ
50 sbsbc a b x a φ [˙a / b]˙ x a φ
51 sbsbc x a φ [˙x / a]˙ φ
52 51 sbcbii [˙a / b]˙ x a φ [˙a / b]˙ [˙x / a]˙ φ
53 50 52 bitri a b x a φ [˙a / b]˙ [˙x / a]˙ φ
54 53 sbcbii [˙b / x]˙ a b x a φ [˙b / x]˙ [˙a / b]˙ [˙x / a]˙ φ
55 49 54 bitri b x a b x a φ [˙b / x]˙ [˙a / b]˙ [˙x / a]˙ φ
56 2sp b a b x a b x a φ φ b x a b x a φ φ
57 55 56 bitr3id b a b x a b x a φ φ [˙b / x]˙ [˙a / b]˙ [˙x / a]˙ φ φ
58 48 57 sylbi b a φ [˙b / x]˙ [˙a / b]˙ [˙x / a]˙ φ φ
59 47 58 sylan9bbr b a φ x = b y = a [˙a / b]˙ [˙x / a]˙ φ φ
60 45 59 bitrd b a φ x = b y = a [˙y / b]˙ [˙x / a]˙ φ φ
61 42 60 anbi12d b a φ x = b y = a A B = x y [˙y / b]˙ [˙x / a]˙ φ A B = b a φ
62 38 39 61 spc2ed b a φ b V a V A B = b a φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
63 34 37 62 sylc A X B X a b φ φ A = b B = a x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
64 63 exp31 A X B X a b φ φ A = b B = a x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
65 64 com23 A X B X a b φ A = b B = a φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
66 27 65 jaod A X B X a b φ A = a B = b A = b B = a φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
67 25 66 sylbid A X B X a b φ A B = a b φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
68 67 impd A X B X a b φ A B = a b φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
69 15 20 68 exlimd A X B X a b φ b A B = a b φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
70 4 11 69 exlimd A X B X a b φ a b A B = a b φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
71 or2expropbilem2 a b A B = a b φ x y A B = x y [˙y / b]˙ [˙x / a]˙ φ
72 70 71 syl6ibr A X B X a b φ a b A B = a b φ a b A B = a b φ
73 oppr A X B X A B = a b A B = a b
74 73 anim1d A X B X A B = a b φ A B = a b φ
75 74 2eximdv A X B X a b A B = a b φ a b A B = a b φ
76 75 3adant3 A X B X a b φ a b A B = a b φ a b A B = a b φ
77 72 76 impbid A X B X a b φ a b A B = a b φ a b A B = a b φ