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Theorem vtocl3gaf 2923
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
vtocl3gaf.a xA
vtocl3gaf.b yA
vtocl3gaf.c zA
vtocl3gaf.d yB
vtocl3gaf.e zB
vtocl3gaf.f zC
vtocl3gaf.1 xψ
vtocl3gaf.2 yχ
vtocl3gaf.3 zθ
vtocl3gaf.4 (x = A → (φψ))
vtocl3gaf.5 (y = B → (ψχ))
vtocl3gaf.6 (z = C → (χθ))
vtocl3gaf.7 ((x R y S z T) → φ)
Assertion
Ref Expression
vtocl3gaf ((A R B S C T) → θ)
Distinct variable groups:   x,y,z,R   x,S,y,z   x,T,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)   χ(x,y,z)   θ(x,y,z)   A(x,y,z)   B(x,y,z)   C(x,y,z)

Proof of Theorem vtocl3gaf
StepHypRef Expression
1 vtocl3gaf.a . . 3 xA
2 vtocl3gaf.b . . 3 yA
3 vtocl3gaf.c . . 3 zA
4 vtocl3gaf.d . . 3 yB
5 vtocl3gaf.e . . 3 zB
6 vtocl3gaf.f . . 3 zC
71nfel1 2499 . . . . 5 x A R
8 nfv 1619 . . . . 5 x y S
9 nfv 1619 . . . . 5 x z T
107, 8, 9nf3an 1827 . . . 4 x(A R y S z T)
11 vtocl3gaf.1 . . . 4 xψ
1210, 11nfim 1813 . . 3 x((A R y S z T) → ψ)
132nfel1 2499 . . . . 5 y A R
144nfel1 2499 . . . . 5 y B S
15 nfv 1619 . . . . 5 y z T
1613, 14, 15nf3an 1827 . . . 4 y(A R B S z T)
17 vtocl3gaf.2 . . . 4 yχ
1816, 17nfim 1813 . . 3 y((A R B S z T) → χ)
193nfel1 2499 . . . . 5 z A R
205nfel1 2499 . . . . 5 z B S
216nfel1 2499 . . . . 5 z C T
2219, 20, 21nf3an 1827 . . . 4 z(A R B S C T)
23 vtocl3gaf.3 . . . 4 zθ
2422, 23nfim 1813 . . 3 z((A R B S C T) → θ)
25 eleq1 2413 . . . . 5 (x = A → (x RA R))
26253anbi1d 1256 . . . 4 (x = A → ((x R y S z T) ↔ (A R y S z T)))
27 vtocl3gaf.4 . . . 4 (x = A → (φψ))
2826, 27imbi12d 311 . . 3 (x = A → (((x R y S z T) → φ) ↔ ((A R y S z T) → ψ)))
29 eleq1 2413 . . . . 5 (y = B → (y SB S))
30293anbi2d 1257 . . . 4 (y = B → ((A R y S z T) ↔ (A R B S z T)))
31 vtocl3gaf.5 . . . 4 (y = B → (ψχ))
3230, 31imbi12d 311 . . 3 (y = B → (((A R y S z T) → ψ) ↔ ((A R B S z T) → χ)))
33 eleq1 2413 . . . . 5 (z = C → (z TC T))
34333anbi3d 1258 . . . 4 (z = C → ((A R B S z T) ↔ (A R B S C T)))
35 vtocl3gaf.6 . . . 4 (z = C → (χθ))
3634, 35imbi12d 311 . . 3 (z = C → (((A R B S z T) → χ) ↔ ((A R B S C T) → θ)))
37 vtocl3gaf.7 . . 3 ((x R y S z T) → φ)
381, 2, 3, 4, 5, 6, 12, 18, 24, 28, 32, 36, 37vtocl3gf 2917 . 2 ((A R B S C T) → ((A R B S C T) → θ))
3938pm2.43i 43 1 ((A R B S C T) → θ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   w3a 934  wnf 1544   = wceq 1642   wcel 1710  wnfc 2476
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-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861
This theorem is referenced by:  vtocl3ga  2924
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