Metamath Proof Explorer


Theorem copsexg

Description: Substitution of class A for ordered pair <. x , y >. . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker copsexgw when possible. (Contributed by NM, 27-Dec-1996) (Revised by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 25-Aug-2019) (New usage is discouraged.)

Ref Expression
Assertion copsexg
|- ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )

Proof

Step Hyp Ref Expression
1 vex
 |-  x e. _V
2 vex
 |-  y e. _V
3 1 2 eqvinop
 |-  ( A = <. x , y >. <-> E. z E. w ( A = <. z , w >. /\ <. z , w >. = <. x , y >. ) )
4 19.8a
 |-  ( E. y ( <. z , w >. = <. x , y >. /\ ph ) -> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) )
5 4 19.23bi
 |-  ( ( <. z , w >. = <. x , y >. /\ ph ) -> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) )
6 5 ex
 |-  ( <. z , w >. = <. x , y >. -> ( ph -> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) ) )
7 vex
 |-  z e. _V
8 vex
 |-  w e. _V
9 7 8 opth
 |-  ( <. z , w >. = <. x , y >. <-> ( z = x /\ w = y ) )
10 9 anbi1i
 |-  ( ( <. z , w >. = <. x , y >. /\ ph ) <-> ( ( z = x /\ w = y ) /\ ph ) )
11 10 2exbii
 |-  ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. x E. y ( ( z = x /\ w = y ) /\ ph ) )
12 nfe1
 |-  F/ x E. x ( z = x /\ E. y ( w = y /\ ph ) )
13 19.8a
 |-  ( ( w = y /\ ph ) -> E. y ( w = y /\ ph ) )
14 13 anim2i
 |-  ( ( z = x /\ ( w = y /\ ph ) ) -> ( z = x /\ E. y ( w = y /\ ph ) ) )
15 14 anassrs
 |-  ( ( ( z = x /\ w = y ) /\ ph ) -> ( z = x /\ E. y ( w = y /\ ph ) ) )
16 15 eximi
 |-  ( E. y ( ( z = x /\ w = y ) /\ ph ) -> E. y ( z = x /\ E. y ( w = y /\ ph ) ) )
17 biidd
 |-  ( A. y y = x -> ( ( z = x /\ E. y ( w = y /\ ph ) ) <-> ( z = x /\ E. y ( w = y /\ ph ) ) ) )
18 17 drex1
 |-  ( A. y y = x -> ( E. y ( z = x /\ E. y ( w = y /\ ph ) ) <-> E. x ( z = x /\ E. y ( w = y /\ ph ) ) ) )
19 16 18 syl5ib
 |-  ( A. y y = x -> ( E. y ( ( z = x /\ w = y ) /\ ph ) -> E. x ( z = x /\ E. y ( w = y /\ ph ) ) ) )
20 anass
 |-  ( ( ( z = x /\ w = y ) /\ ph ) <-> ( z = x /\ ( w = y /\ ph ) ) )
21 20 exbii
 |-  ( E. y ( ( z = x /\ w = y ) /\ ph ) <-> E. y ( z = x /\ ( w = y /\ ph ) ) )
22 19.40
 |-  ( E. y ( z = x /\ ( w = y /\ ph ) ) -> ( E. y z = x /\ E. y ( w = y /\ ph ) ) )
23 nfeqf2
 |-  ( -. A. y y = x -> F/ y z = x )
24 23 19.9d
 |-  ( -. A. y y = x -> ( E. y z = x -> z = x ) )
25 24 anim1d
 |-  ( -. A. y y = x -> ( ( E. y z = x /\ E. y ( w = y /\ ph ) ) -> ( z = x /\ E. y ( w = y /\ ph ) ) ) )
26 22 25 syl5
 |-  ( -. A. y y = x -> ( E. y ( z = x /\ ( w = y /\ ph ) ) -> ( z = x /\ E. y ( w = y /\ ph ) ) ) )
27 21 26 syl5bi
 |-  ( -. A. y y = x -> ( E. y ( ( z = x /\ w = y ) /\ ph ) -> ( z = x /\ E. y ( w = y /\ ph ) ) ) )
28 19.8a
 |-  ( ( z = x /\ E. y ( w = y /\ ph ) ) -> E. x ( z = x /\ E. y ( w = y /\ ph ) ) )
29 27 28 syl6
 |-  ( -. A. y y = x -> ( E. y ( ( z = x /\ w = y ) /\ ph ) -> E. x ( z = x /\ E. y ( w = y /\ ph ) ) ) )
30 19 29 pm2.61i
 |-  ( E. y ( ( z = x /\ w = y ) /\ ph ) -> E. x ( z = x /\ E. y ( w = y /\ ph ) ) )
31 12 30 exlimi
 |-  ( E. x E. y ( ( z = x /\ w = y ) /\ ph ) -> E. x ( z = x /\ E. y ( w = y /\ ph ) ) )
32 euequ
 |-  E! x x = z
33 equcom
 |-  ( x = z <-> z = x )
34 33 eubii
 |-  ( E! x x = z <-> E! x z = x )
35 32 34 mpbi
 |-  E! x z = x
36 eupick
 |-  ( ( E! x z = x /\ E. x ( z = x /\ E. y ( w = y /\ ph ) ) ) -> ( z = x -> E. y ( w = y /\ ph ) ) )
37 35 36 mpan
 |-  ( E. x ( z = x /\ E. y ( w = y /\ ph ) ) -> ( z = x -> E. y ( w = y /\ ph ) ) )
38 37 com12
 |-  ( z = x -> ( E. x ( z = x /\ E. y ( w = y /\ ph ) ) -> E. y ( w = y /\ ph ) ) )
39 euequ
 |-  E! y y = w
40 equcom
 |-  ( y = w <-> w = y )
41 40 eubii
 |-  ( E! y y = w <-> E! y w = y )
42 39 41 mpbi
 |-  E! y w = y
43 eupick
 |-  ( ( E! y w = y /\ E. y ( w = y /\ ph ) ) -> ( w = y -> ph ) )
44 42 43 mpan
 |-  ( E. y ( w = y /\ ph ) -> ( w = y -> ph ) )
45 44 com12
 |-  ( w = y -> ( E. y ( w = y /\ ph ) -> ph ) )
46 38 45 sylan9
 |-  ( ( z = x /\ w = y ) -> ( E. x ( z = x /\ E. y ( w = y /\ ph ) ) -> ph ) )
47 31 46 syl5
 |-  ( ( z = x /\ w = y ) -> ( E. x E. y ( ( z = x /\ w = y ) /\ ph ) -> ph ) )
48 11 47 syl5bi
 |-  ( ( z = x /\ w = y ) -> ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) -> ph ) )
49 9 48 sylbi
 |-  ( <. z , w >. = <. x , y >. -> ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) -> ph ) )
50 6 49 impbid
 |-  ( <. z , w >. = <. x , y >. -> ( ph <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) ) )
51 eqeq1
 |-  ( A = <. z , w >. -> ( A = <. x , y >. <-> <. z , w >. = <. x , y >. ) )
52 51 anbi1d
 |-  ( A = <. z , w >. -> ( ( A = <. x , y >. /\ ph ) <-> ( <. z , w >. = <. x , y >. /\ ph ) ) )
53 52 2exbidv
 |-  ( A = <. z , w >. -> ( E. x E. y ( A = <. x , y >. /\ ph ) <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) ) )
54 53 bibi2d
 |-  ( A = <. z , w >. -> ( ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) <-> ( ph <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) ) ) )
55 51 54 imbi12d
 |-  ( A = <. z , w >. -> ( ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) <-> ( <. z , w >. = <. x , y >. -> ( ph <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) ) ) ) )
56 50 55 mpbiri
 |-  ( A = <. z , w >. -> ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) )
57 56 adantr
 |-  ( ( A = <. z , w >. /\ <. z , w >. = <. x , y >. ) -> ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) )
58 57 exlimivv
 |-  ( E. z E. w ( A = <. z , w >. /\ <. z , w >. = <. x , y >. ) -> ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) )
59 3 58 sylbi
 |-  ( A = <. x , y >. -> ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) ) )
60 59 pm2.43i
 |-  ( A = <. x , y >. -> ( ph <-> E. x E. y ( A = <. x , y >. /\ ph ) ) )