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Theorem vtocl2gaf 2921
 Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
vtocl2gaf.a xA
vtocl2gaf.b yA
vtocl2gaf.c yB
vtocl2gaf.1 xψ
vtocl2gaf.2 yχ
vtocl2gaf.3 (x = A → (φψ))
vtocl2gaf.4 (y = B → (ψχ))
vtocl2gaf.5 ((x C y D) → φ)
Assertion
Ref Expression
vtocl2gaf ((A C B D) → χ)
Distinct variable groups:   x,y,C   x,D,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   χ(x,y)   A(x,y)   B(x,y)

Proof of Theorem vtocl2gaf
StepHypRef Expression
1 vtocl2gaf.a . . 3 xA
2 vtocl2gaf.b . . 3 yA
3 vtocl2gaf.c . . 3 yB
41nfel1 2499 . . . . 5 x A C
5 nfv 1619 . . . . 5 x y D
64, 5nfan 1824 . . . 4 x(A C y D)
7 vtocl2gaf.1 . . . 4 xψ
86, 7nfim 1813 . . 3 x((A C y D) → ψ)
92nfel1 2499 . . . . 5 y A C
103nfel1 2499 . . . . 5 y B D
119, 10nfan 1824 . . . 4 y(A C B D)
12 vtocl2gaf.2 . . . 4 yχ
1311, 12nfim 1813 . . 3 y((A C B D) → χ)
14 eleq1 2413 . . . . 5 (x = A → (x CA C))
1514anbi1d 685 . . . 4 (x = A → ((x C y D) ↔ (A C y D)))
16 vtocl2gaf.3 . . . 4 (x = A → (φψ))
1715, 16imbi12d 311 . . 3 (x = A → (((x C y D) → φ) ↔ ((A C y D) → ψ)))
18 eleq1 2413 . . . . 5 (y = B → (y DB D))
1918anbi2d 684 . . . 4 (y = B → ((A C y D) ↔ (A C B D)))
20 vtocl2gaf.4 . . . 4 (y = B → (ψχ))
2119, 20imbi12d 311 . . 3 (y = B → (((A C y D) → ψ) ↔ ((A C B D) → χ)))
22 vtocl2gaf.5 . . 3 ((x C y D) → φ)
231, 2, 3, 8, 13, 17, 21, 22vtocl2gf 2916 . 2 ((A C B D) → ((A C B D) → χ))
2423pm2.43i 43 1 ((A C B D) → χ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 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-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:  vtocl2ga  2922  ov3  5599  ov2gf  5711
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