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Theorem cgsex4g 2892
Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
Hypotheses
Ref Expression
cgsex4g.1 (((x = A y = B) (z = C w = D)) → χ)
cgsex4g.2 (χ → (φψ))
Assertion
Ref Expression
cgsex4g (((A R B S) (C R D S)) → (xyzw(χ φ) ↔ ψ))
Distinct variable groups:   x,y,z,w,A   x,B,y,z,w   x,C,y,z,w   x,D,y,z,w   ψ,x,y,z,w
Allowed substitution hints:   φ(x,y,z,w)   χ(x,y,z,w)   R(x,y,z,w)   S(x,y,z,w)

Proof of Theorem cgsex4g
StepHypRef Expression
1 cgsex4g.2 . . . . 5 (χ → (φψ))
21biimpa 470 . . . 4 ((χ φ) → ψ)
32exlimivv 1635 . . 3 (zw(χ φ) → ψ)
43exlimivv 1635 . 2 (xyzw(χ φ) → ψ)
5 elisset 2869 . . . . . . . 8 (A Rx x = A)
6 elisset 2869 . . . . . . . 8 (B Sy y = B)
75, 6anim12i 549 . . . . . . 7 ((A R B S) → (x x = A y y = B))
8 eeanv 1913 . . . . . . 7 (xy(x = A y = B) ↔ (x x = A y y = B))
97, 8sylibr 203 . . . . . 6 ((A R B S) → xy(x = A y = B))
10 elisset 2869 . . . . . . . 8 (C Rz z = C)
11 elisset 2869 . . . . . . . 8 (D Sw w = D)
1210, 11anim12i 549 . . . . . . 7 ((C R D S) → (z z = C w w = D))
13 eeanv 1913 . . . . . . 7 (zw(z = C w = D) ↔ (z z = C w w = D))
1412, 13sylibr 203 . . . . . 6 ((C R D S) → zw(z = C w = D))
159, 14anim12i 549 . . . . 5 (((A R B S) (C R D S)) → (xy(x = A y = B) zw(z = C w = D)))
16 ee4anv 1915 . . . . 5 (xyzw((x = A y = B) (z = C w = D)) ↔ (xy(x = A y = B) zw(z = C w = D)))
1715, 16sylibr 203 . . . 4 (((A R B S) (C R D S)) → xyzw((x = A y = B) (z = C w = D)))
18 cgsex4g.1 . . . . . 6 (((x = A y = B) (z = C w = D)) → χ)
19182eximi 1577 . . . . 5 (zw((x = A y = B) (z = C w = D)) → zwχ)
20192eximi 1577 . . . 4 (xyzw((x = A y = B) (z = C w = D)) → xyzwχ)
2117, 20syl 15 . . 3 (((A R B S) (C R D S)) → xyzwχ)
221biimprcd 216 . . . . . 6 (ψ → (χφ))
2322ancld 536 . . . . 5 (ψ → (χ → (χ φ)))
24232eximdv 1624 . . . 4 (ψ → (zwχzw(χ φ)))
25242eximdv 1624 . . 3 (ψ → (xyzwχxyzw(χ φ)))
2621, 25syl5com 26 . 2 (((A R B S) (C R D S)) → (ψxyzw(χ φ)))
274, 26impbid2 195 1 (((A R B S) (C R D S)) → (xyzw(χ φ) ↔ ψ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358  wex 1541   = wceq 1642   wcel 1710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861
This theorem is referenced by:  copsex4g  4610
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