Metamath Proof Explorer


Theorem cgsex4gOLD

Description: Obsolete version of cgsex4g as of 21-Mar-2025. (Contributed by NM, 5-Aug-1995) Avoid ax-10 , ax-11 . (Revised by Gino Giotto, 28-Jun-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses cgsex4gOLD.1 x = A y = B z = C w = D χ
cgsex4gOLD.2 χ φ ψ
Assertion cgsex4gOLD A R B S C R D S x y z w χ φ ψ

Proof

Step Hyp Ref Expression
1 cgsex4gOLD.1 x = A y = B z = C w = D χ
2 cgsex4gOLD.2 χ φ ψ
3 2 biimpa χ φ ψ
4 3 exlimivv z w χ φ ψ
5 4 exlimivv x y z w χ φ ψ
6 elisset A R x x = A
7 elisset B S y y = B
8 6 7 anim12i A R B S x x = A y y = B
9 exdistrv x y x = A y = B x x = A y y = B
10 8 9 sylibr A R B S x y x = A y = B
11 elisset C R z z = C
12 elisset D S w w = D
13 11 12 anim12i C R D S z z = C w w = D
14 exdistrv z w z = C w = D z z = C w w = D
15 13 14 sylibr C R D S z w z = C w = D
16 10 15 anim12i A R B S C R D S x y x = A y = B z w z = C w = D
17 eqeq1 y = v y = B v = B
18 17 anbi2d y = v x = A y = B x = A v = B
19 18 anbi1d y = v x = A y = B z = C w = D x = A v = B z = C w = D
20 19 exbidv y = v w x = A y = B z = C w = D w x = A v = B z = C w = D
21 20 notbid y = v ¬ w x = A y = B z = C w = D ¬ w x = A v = B z = C w = D
22 eqeq1 z = v z = C v = C
23 22 anbi1d z = v z = C w = D v = C w = D
24 23 anbi2d z = v x = A y = B z = C w = D x = A y = B v = C w = D
25 24 exbidv z = v w x = A y = B z = C w = D w x = A y = B v = C w = D
26 25 notbid z = v ¬ w x = A y = B z = C w = D ¬ w x = A y = B v = C w = D
27 21 26 alcomw y z ¬ w x = A y = B z = C w = D z y ¬ w x = A y = B z = C w = D
28 27 notbii ¬ y z ¬ w x = A y = B z = C w = D ¬ z y ¬ w x = A y = B z = C w = D
29 2exnaln y z w x = A y = B z = C w = D ¬ y z ¬ w x = A y = B z = C w = D
30 2exnaln z y w x = A y = B z = C w = D ¬ z y ¬ w x = A y = B z = C w = D
31 28 29 30 3bitr4i y z w x = A y = B z = C w = D z y w x = A y = B z = C w = D
32 31 exbii x y z w x = A y = B z = C w = D x z y w x = A y = B z = C w = D
33 4exdistrv x z y w x = A y = B z = C w = D x y x = A y = B z w z = C w = D
34 32 33 bitri x y z w x = A y = B z = C w = D x y x = A y = B z w z = C w = D
35 16 34 sylibr A R B S C R D S x y z w x = A y = B z = C w = D
36 1 2eximi z w x = A y = B z = C w = D z w χ
37 36 2eximi x y z w x = A y = B z = C w = D x y z w χ
38 35 37 syl A R B S C R D S x y z w χ
39 2 biimprcd ψ χ φ
40 39 ancld ψ χ χ φ
41 40 2eximdv ψ z w χ z w χ φ
42 41 2eximdv ψ x y z w χ x y z w χ φ
43 38 42 syl5com A R B S C R D S ψ x y z w χ φ
44 5 43 impbid2 A R B S C R D S x y z w χ φ ψ