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Theorem cgsex4g 2585
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 280 . . . 4 ((χ φ) → ψ)
32exlimivv 1773 . . 3 (zw(χ φ) → ψ)
43exlimivv 1773 . 2 (xyzw(χ φ) → ψ)
5 elisset 2562 . . . . . . . 8 (A 𝑅x x = A)
6 elisset 2562 . . . . . . . 8 (B 𝑆y y = B)
75, 6anim12i 321 . . . . . . 7 ((A 𝑅 B 𝑆) → (x x = A y y = B))
8 eeanv 1804 . . . . . . 7 (xy(x = A y = B) ↔ (x x = A y y = B))
97, 8sylibr 137 . . . . . 6 ((A 𝑅 B 𝑆) → xy(x = A y = B))
10 elisset 2562 . . . . . . . 8 (𝐶 𝑅z z = 𝐶)
11 elisset 2562 . . . . . . . 8 (𝐷 𝑆w w = 𝐷)
1210, 11anim12i 321 . . . . . . 7 ((𝐶 𝑅 𝐷 𝑆) → (z z = 𝐶 w w = 𝐷))
13 eeanv 1804 . . . . . . 7 (zw(z = 𝐶 w = 𝐷) ↔ (z z = 𝐶 w w = 𝐷))
1412, 13sylibr 137 . . . . . 6 ((𝐶 𝑅 𝐷 𝑆) → zw(z = 𝐶 w = 𝐷))
159, 14anim12i 321 . . . . 5 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xy(x = A y = B) zw(z = 𝐶 w = 𝐷)))
16 ee4anv 1806 . . . . 5 (xyzw((x = A y = B) (z = 𝐶 w = 𝐷)) ↔ (xy(x = A y = B) zw(z = 𝐶 w = 𝐷)))
1715, 16sylibr 137 . . . 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 1489 . . . . 5 (zw((x = A y = B) (z = 𝐶 w = 𝐷)) → zwχ)
20192eximi 1489 . . . 4 (xyzw((x = A y = B) (z = 𝐶 w = 𝐷)) → xyzwχ)
2117, 20syl 14 . . 3 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → xyzwχ)
221biimprcd 149 . . . . . 6 (ψ → (χφ))
2322ancld 308 . . . . 5 (ψ → (χ → (χ φ)))
24232eximdv 1759 . . . 4 (ψ → (zwχzw(χ φ)))
25242eximdv 1759 . . 3 (ψ → (xyzwχxyzw(χ φ)))
2621, 25syl5com 26 . 2 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (ψxyzw(χ φ)))
274, 26impbid2 131 1 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xyzw(χ φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  copsex4g  3975  brecop  6132
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