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Theorem vtocl3ga 2924
 Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtocl3ga.1 (x = A → (φψ))
vtocl3ga.2 (y = B → (ψχ))
vtocl3ga.3 (z = C → (χθ))
vtocl3ga.4 ((x D y R z S) → φ)
Assertion
Ref Expression
vtocl3ga ((A D B R C S) → θ)
Distinct variable groups:   x,y,z,A   y,B,z   z,C   x,D,y,z   x,R,y,z   x,S,y,z   ψ,x   χ,y   θ,z
Allowed substitution hints:   φ(x,y,z)   ψ(y,z)   χ(x,z)   θ(x,y)   B(x)   C(x,y)

Proof of Theorem vtocl3ga
StepHypRef Expression
1 nfcv 2489 . 2 xA
2 nfcv 2489 . 2 yA
3 nfcv 2489 . 2 zA
4 nfcv 2489 . 2 yB
5 nfcv 2489 . 2 zB
6 nfcv 2489 . 2 zC
7 nfv 1619 . 2 xψ
8 nfv 1619 . 2 yχ
9 nfv 1619 . 2 zθ
10 vtocl3ga.1 . 2 (x = A → (φψ))
11 vtocl3ga.2 . 2 (y = B → (ψχ))
12 vtocl3ga.3 . 2 (z = C → (χθ))
13 vtocl3ga.4 . 2 ((x D y R z S) → φ)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13vtocl3gaf 2923 1 ((A D B R C S) → θ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  preq12bg  4128
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