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Theorem vtocl2gf 2916
 Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.)
Hypotheses
Ref Expression
vtocl2gf.1 xA
vtocl2gf.2 yA
vtocl2gf.3 yB
vtocl2gf.4 xψ
vtocl2gf.5 yχ
vtocl2gf.6 (x = A → (φψ))
vtocl2gf.7 (y = B → (ψχ))
vtocl2gf.8 φ
Assertion
Ref Expression
vtocl2gf ((A V B W) → χ)

Proof of Theorem vtocl2gf
StepHypRef Expression
1 elex 2867 . 2 (A VA V)
2 vtocl2gf.3 . . 3 yB
3 vtocl2gf.2 . . . . 5 yA
43nfel1 2499 . . . 4 y A V
5 vtocl2gf.5 . . . 4 yχ
64, 5nfim 1813 . . 3 y(A V → χ)
7 vtocl2gf.7 . . . 4 (y = B → (ψχ))
87imbi2d 307 . . 3 (y = B → ((A V → ψ) ↔ (A V → χ)))
9 vtocl2gf.1 . . . 4 xA
10 vtocl2gf.4 . . . 4 xψ
11 vtocl2gf.6 . . . 4 (x = A → (φψ))
12 vtocl2gf.8 . . . 4 φ
139, 10, 11, 12vtoclgf 2913 . . 3 (A V → ψ)
142, 6, 8, 13vtoclgf 2913 . 2 (B W → (A V → χ))
151, 14mpan9 455 1 ((A V B W) → χ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  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-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:  vtocl3gf  2917  vtocl2g  2918  vtocl2gaf  2921
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