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Theorem vtocl2gf 2583
 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 2534 . 2 (A 𝑉A V)
2 vtocl2gf.3 . . 3 yB
3 vtocl2gf.2 . . . . 5 yA
43nfel1 2161 . . . 4 y A V
5 vtocl2gf.5 . . . 4 yχ
64, 5nfim 1437 . . 3 y(A V → χ)
7 vtocl2gf.7 . . . 4 (y = B → (ψχ))
87imbi2d 219 . . 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 2580 . . 3 (A V → ψ)
142, 6, 8, 13vtoclgf 2580 . 2 (B 𝑊 → (A V → χ))
151, 14mpan9 265 1 ((A 𝑉 B 𝑊) → χ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1223  Ⅎwnf 1322   ∈ wcel 1366  Ⅎwnfc 2138  Vcvv 2526 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995 This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528 This theorem is referenced by:  vtocl3gf  2584  vtocl2g  2585  vtocl2gaf  2588
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