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Theorem vtocl3gf 2917
 Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtocl3gf.a xA
vtocl3gf.b yA
vtocl3gf.c zA
vtocl3gf.d yB
vtocl3gf.e zB
vtocl3gf.f zC
vtocl3gf.1 xψ
vtocl3gf.2 yχ
vtocl3gf.3 zθ
vtocl3gf.4 (x = A → (φψ))
vtocl3gf.5 (y = B → (ψχ))
vtocl3gf.6 (z = C → (χθ))
vtocl3gf.7 φ
Assertion
Ref Expression
vtocl3gf ((A V B W C X) → θ)

Proof of Theorem vtocl3gf
StepHypRef Expression
1 elex 2867 . . 3 (A VA V)
2 vtocl3gf.d . . . 4 yB
3 vtocl3gf.e . . . 4 zB
4 vtocl3gf.f . . . 4 zC
5 vtocl3gf.b . . . . . 6 yA
65nfel1 2499 . . . . 5 y A V
7 vtocl3gf.2 . . . . 5 yχ
86, 7nfim 1813 . . . 4 y(A V → χ)
9 vtocl3gf.c . . . . . 6 zA
109nfel1 2499 . . . . 5 z A V
11 vtocl3gf.3 . . . . 5 zθ
1210, 11nfim 1813 . . . 4 z(A V → θ)
13 vtocl3gf.5 . . . . 5 (y = B → (ψχ))
1413imbi2d 307 . . . 4 (y = B → ((A V → ψ) ↔ (A V → χ)))
15 vtocl3gf.6 . . . . 5 (z = C → (χθ))
1615imbi2d 307 . . . 4 (z = C → ((A V → χ) ↔ (A V → θ)))
17 vtocl3gf.a . . . . 5 xA
18 vtocl3gf.1 . . . . 5 xψ
19 vtocl3gf.4 . . . . 5 (x = A → (φψ))
20 vtocl3gf.7 . . . . 5 φ
2117, 18, 19, 20vtoclgf 2913 . . . 4 (A V → ψ)
222, 3, 4, 8, 12, 14, 16, 21vtocl2gf 2916 . . 3 ((B W C X) → (A V → θ))
231, 22mpan9 455 . 2 ((A V (B W C X)) → θ)
24233impb 1147 1 ((A V B W C X) → θ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859 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:  vtocl3gaf  2923
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