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Theorem vtocldf 2906
 Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1 (φA V)
vtocld.2 ((φ x = A) → (ψχ))
vtocld.3 (φψ)
vtocldf.4 xφ
vtocldf.5 (φxA)
vtocldf.6 (φ → Ⅎxχ)
Assertion
Ref Expression
vtocldf (φχ)

Proof of Theorem vtocldf
StepHypRef Expression
1 vtocldf.5 . 2 (φxA)
2 vtocldf.6 . 2 (φ → Ⅎxχ)
3 vtocldf.4 . . 3 xφ
4 vtocld.2 . . . 4 ((φ x = A) → (ψχ))
54ex 423 . . 3 (φ → (x = A → (ψχ)))
63, 5alrimi 1765 . 2 (φx(x = A → (ψχ)))
7 vtocld.3 . . 3 (φψ)
83, 7alrimi 1765 . 2 (φxψ)
9 vtocld.1 . 2 (φA V)
10 vtoclgft 2905 . 2 (((xA xχ) (x(x = A → (ψχ)) xψ) A V) → χ)
111, 2, 6, 8, 9, 10syl221anc 1193 1 (φχ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  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-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:  vtocld  2907
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